Spatio-temporal coherence of free electron laser pulses in the soft x-ray regime

The temporal coherence properties of soft x-ray free electron laser pulses at FLASH are measured at 23.9 nm by interfering two timedelayed partial beams directly on a CCD camera. The partial beams are obtained by wave front beam splitting in an autocorrelator operating at photon energies from hν = 30 to 200 eV. At zero delay a visibility of (0.63 ± 0.04) is measured. The delay of one partial beam reveals a coherence time of 6 fs at 23.9 nm. The visibility further displays a non-monotonic decay, which can be rationalized by the presence of multiple pulse structure. 2008 Optical Society of America OCIS codes: (030.1640) Coherence; (140.2600) Free-electron lasers (FELs); (340.7450) X-ray interferometry; (340.7480) X-rays, soft x-rays, extreme ultraviolet (EUV) References and links 1. H. Wabnitz, L. Bittner, A. R. B. de Castro, R. Döhrmann, P. Gürtler, T. Laarmann, W. Laasch, J. Schulz, A. Swiderski, K. von Haeften, T. Möller, B. Faatz, A. Fateev, J. Feldhaus, C. Gerth, U. Hahn, E. 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Introduction
In the soft x-ray spectral regime free electron laser (FEL) sources provide ultrashort radiation pulses with unprecedented intensity and brilliance.Great efforts are devoted to apply such sources for the investigation of fundamental questions about ultrafast dynamical processes in biological, chemical and physical systems.First basic results in cluster physics [1,2] and single pulse holographic imaging [3,4] have been obtained at the free electron laser at Hamburg (FLASH) which operates down to 6 nm [5].For the interpretation of experimental data and image retrieval of dynamic processes a precise knowledge of the phase coherence properties of the FEL pulses and their temporal profile is of utmost importance.Femtosecond time-resolved investigations in the x-ray spectral regime of, e.g., multiphoton processes in atoms [6], cluster dynamics, and the dynamic behaviour of magnetic nanostructures [7], will advance the understanding of ultrafast processes in basic and applied sciences.
For an experiment free of time jitter in the low femtosecond regime both pulses are derived from the same optical source by intensity or wavefront beam splitting.Intensity and amplitude correlation techniques or time-delayed pulses with the same or different colour are then applied to derive the information about the system under study.Free electron laser x-ray sources provide unique and attractive light pulses with expected pulse durations in the low femtosecond range.
While FEL radiation in the visible and near-UV spectral range has been studied in detail [8] the coherence properties and pulse durations of soft x-ray FEL pulses have not yet been characterized.In the visible, phase sensitive techniques, like FROG [9], have been employed to characterize the properties of FEL radiation emitted from long electron bunches and compare the data to theory [8].Such sophisticated methods are still waiting to be developed for the VUV and soft x-ray spectral range.Nevertheless, Ischebeck et al. [10] determined the spatial coherence of a VUV FEL at photon wavelengths between 80 and 120 nm by a Young's double slit experiment.Furthermore, indirect information on the temporal coherence has been estimated from single shot spectral analysis [11,12] and a statistical evaluation of the intensity fluctuations [5,8,13].
The theoretical description of the FEL process based on self-amplified spontaneous emission (SASE) and the built-up of coherence has made tremendous progress in the recent past, see e.g.[14,15,16].However, in particular for the femtosecond mode in which FLASH is operating still open questions remain also in the theoretical description.The direct measurement of the temporal coherence of soft x-ray FEL pulses can deliver valuable input for a theoretical understanding.It is therefore the purpose of the present study to extend the knowledge about coherence properties of SASE FEL radiation into the soft x-ray regime for the first time.Further, the radiation of an existing FEL source operated as user facility is characterized, providing basic information for the design of experiments for a wide-spread user community.Knowledge of the variation of these parameters across the spatial beam profile allows to take full advantage of the radiation provided.

Experimental set-up
A beam splitter and pulse delay unit has been especially developed for the use with FLASH [17] to provide jitter-free replica pulses for x-ray pump/x-ray probe experiments (Fig. 1).Due to the special optical requirements in the soft x-ray regime wave front splitting devices and grazing incidence mirrors are best suited to cover a broad photon energy range with high transmission.Wave front beam splitting always causes diffraction from the edge of the beam splitter giving rise to unwanted modulations of the split beam.However, these modulations have a much lower spatial frequency than the interference fringes due to the overlapped split beams in the present linear autocorrelation experiments.Figure 1(b).shows the calculated transmission in both arms over the photon energy range from 30 eV to 200 eV (λ = 41 to 6.2 nm).After four reflections from commercial amorphous carbon coated Si mirrors a total transmission of more than 65 % is expected for the fixed arm and more than 50% (up to hν 160 eV) for the variable arm, decreasing at higher photon energies.A three dimensional beam path has been chosen with perpendicular arranged pathways for the fixed and variable delay beams, see figure 1(a).The optical concept is based on wave front beam splitting and grazing incidence geometry at the mirrors.In detail, the left part of the incoming FEL beam is horizontally reflected in x-direction by a mirror with a sharp edge, i.e. the beam splitter, into the fixed delay line.The unaffected right part of the beam is vertically reflected in y-direction by the subsequent mirror onto the two mirrors of the variable delay stage.Eventually, the half beams (both carrying a defined delay) are reflected into their original direction.The incidence angles in the fixed delay arm and in the variable delay arm are 3° and 6°, respectively, as measured from the plane of the mirrors.The mechanical movement of the mirrors in the variable arm allows a time delay of the pulse travelling this path from -3 ps to + 20 ps with respect to pulses in the fixed arm.Driven by a stepping motor with gear box the nominal step size is about 0.04 fs.In practice sub-fs resolution is achieved under stable thermal conditions.The mirror assemblies offer true gimbal adjustments with an angular resolution of 0.3 µrad and a useful range of 10 mrad.The free electron laser FLASH was running in the so-called femtosecond mode in which the electron bunch is compressed to a very short spike followed by a longer tail [15].The duration of the spike is expected to be as short as 50 fs.Due to experimental constraints only an upper limit of 120 fs spike duration could be determined experimentally [13].During the present measurements an average bunch charge of 0.6 nC at an energy of 526 MeV was recorded, yielding saturated FEL radiation at 23.5 to 24 nm with pulse energies between 11 µJ and 15 µJ (σ < 5 %).The beam size at the experiment was about 8 mm (FWHM) in diameter both in x (horizontal) and y (vertical) directions.Further parameters are listed in Table I [18].

Method
The coherence of an optical wave field between two positions r1 and r2 and for two times t and (t + τ) is described by the mutual correlation function [19] This simplifies for a fixed separation (r1 -r2) to the longitudinal coherence or mutual time correlation function The visibility V yields an experimentally easily accessible value for the normalized mutual correlation function where Imax and Imin are the maximal and minimal intensities of the observed fringes.Here also different intensities of the two interfering beams can be considered.In the present experiment both partial beams brought to interference show the same intensities in the overlap area.
For the coherence measurements both partial beams are brought to an overlap at a distance of L = 10 m from the last mirror.For the quantitative evaluation of the temporal coherence a crossing angle of α = 0.18 mrad is chosen resulting in about six interference fringes with a separation of about δ = 130 µm for λ = 23.9nm, see below.This large fringe separation is chosen in order to avoid a degradation of the visibility by the measurement process, because the pixel size of the soft x-ray CCD camera employed amounted to 13.5 µm.Nevertheless, this still results in a reduction of the true visibility V to an effective visibility Veff = k V by k = 0.96.For smaller fringe spacing this factor k decreases and thus reduces the effective detected visibility further.The visibility has been determined for a single exposure averaging vertically over part of the beam profile, but horizontally only over two fringe maxima.Thus only a minimal additional time uncertainty of 0.08 fs is added which corresponds to the path length difference of one wavelength between two fringes.

Results and discussion
The spatio-temporal coherence of the FEL pulses is determined from two-beam interference pattern generated by overlapping both partial beams under a small angle directly on a soft xray sensitive CCD camera.Interference fringes have been measured for different horizontal overlap adjustable in the autocorrelator.Figure 2 shows examples of single shot exposures for four different crossing angles from α = 0.18 mrad to 0.75 mrad.The fringe separation δ on the CCD camera changes thereby from 130 µm to 32 µm, as expected for a wavelength of 23.9 nm.On all images distinct speckles are observed resulting from a gold mirror which had to be inserted in the beam line behind the autocorrelator.Changing the time delay between both partial beams the temporal coherence properties of the FEL pulses are measured.For this measurement the central 0.80 mm of the FEL beam are overlapped.Figure 3(c).shows the visibility observed as a function of time delay between the two partial beams, averaged over the vertical pulse profile.Each data point (red dots) is the average of the visibility of ten single exposure interference pictures.The (averaged) visibility of V = 0.63 at zero time delay rapidly decreases as the time delay is increased.For single pulses well-contrasted fringes can be discerned even at delay times of 60 to 80 fs, however, after averaging independent pulses only a low visibility is retained.Such strong shot-to-shot fluctuations are only observed at long delay times and may reflect the volatile intensity in the tail of the pulse.The central maximum of the correlation can be described by a Gaussian function (green line) with a width of 12 fs (FWHM).A coherence time, corresponding to half of the full width, of τcoh = 6 fs (or τcoh(rms) = 5.1 fs) is thus arrived at.The corresponding coherence length lcoh = τcoh c is then lcoh = 1.8 µm or about 75 optical cycles of the soft x-ray radiation.A single mode of the FEL pulse is expected to show a spectral width of about 0.7 % of the central wavelength [18].At λ = 23.9nm this width then amounts to about ∆λ = 0.17 nm.Under the assumption of a Fourier transform limited Gaussian pulse this spectral bandwidth can be translated into a coherence length of [19] ( ) ( ) Taking the specifics of the present experiment a value of lcoh = 2.25 µm or τcoh = 7.5 fs is calculated for λ = 24 nm, in very good agreement with the fit to the central experimental peak of the visibility function.Remarkably, the visibility, i.e. the mutual coherence, is not a monotonic function of the delay time between both partial beams.Instead, a minimum at about 7.4 fs after the main maximum and a secondary maximum at about 12.3 fs appear, symmetrically on both sides of the main maximum.In addition, a discernible increase of the visibility occurs at a delay around 40 fs.Since the interferences are measured for independent single pulses of the FEL and then their visibilities are averaged, this behaviour of the temporal coherence function must reflect intrinsic features of the FEL pulses.
It may be illustrative to connect the measured temporal coherence with estimations from FEL theory by a simplified look on the formation of the radiation field and the built-up of coherence in a SASE FEL.The radiation starts from shot noise of the electron bunch.There a multitude of transverse and longitudinal modes arise which are further amplified in the FEL.In the so-called high-gain linear regime the number of modes is depleted and spatial and temporal coherence grows with the FEL acting as a linear filter.For long undulators the gain saturates and the fluctuations are dramatically reduced.This regime is the normal mode of operation for experiments.The high-gain linear mode can be well understood in terms of statistical optics [14,16].In this way estimates of the number of modes as well as their longitudinal length have been derived [18].The cooperation length Lc of the electrons is about with the FEL wavelength λ and the undulator period λU.Assuming a gain length of Lg = 0.8 m (see table I) a cooperation length of Lc = 1.6 µm is estimated at λ = 23.9nm, equivalent to a cooperation time of about 5 fs.This is consistent with the coherence time determined from the visibility curve in Fig. 3. Closely related to the cooperation length is the number of independent temporal modes.For a flat electron bunch distribution this number can roughly be assumed to be the ratio of the electron bunch length to the cooperation time.For FLASH operating in the femtosecond mode, however, the bunch distribution is not flat and therefore this relation does not hold.On the other hand, the average number of modes has recently been determined experimentally from the statistical fluctuations of the pulse energy at λ = 30 nm to about M = 4.1 [20].From this result the authors expect three to four longitudinal modes in each pulse, corresponding to an equal number of spectral modes.Figure 4 displays a selection of single shot spectra recorded during the same shift of FLASH in which the interference pictures are taken.Here, between two and four spectral modes appear, suggesting a similar number of temporal modes.The visibility function of Fig. 3c represents the Fourier transform of the average spectral distribution of the coherent part of the FEL pulses.Since the spectral phase variation of the pulses is not known, the pulse shape can not directly be reconstructed from this measurement without further information.Experimental single shot spectra, independently obtained during these measurements, show multiple spectral modes and overall a significantly larger spectral bandwidth, see Fig. 4 and the discussion above.Therefore, from the spectra a much shorter temporal coherence may mistakenly be deduced, when a fixed phase correlation between the spectral components is assumed, than is directly measured (Fig. 3).Further, both experiments and simulations proof that the FEL radiation consists of several independent modes [5,10,20].Thus the linear autocorrelation is a superposition of the Fourier transforms of such single modes.Even though there is no information on the phase and thus on the temporal shape of a single mode, the temporal distance of independent modes is inherent and can be recognized in the linear autocorrelation.The best agreement with the experimental visibility curve is obtained for a mode sequence consisting of two main and a third trailing mode, as shown in figure 5 (blue line).The three pulses are assumed to have a Gaussian bandwidth of ∆λ = 0.17 nm at 24 nm central wavelength.They are separated by 12 fs and 40 fs with respect to the first pulse with relative respective intensities of 1.0, 0.80 and 0.20, see inset. Figure 5 shows also simulations of the visibility for a pulse separation of 10 fs (red line) and 20 fs (grey line).It is evident that these pulse structures do not fit the observed temporal lapse of the visibility function.For simplicity the pulses displayed in the inset of Fig. 5 are assumed to have a flat phase.Nevertheless, the real pulses are expected to carry a chirp due to the energy chirp of the electron bunch [5], but this does not influence the linear autocorrelation, i.e. the temporal visibility.The central photon energy of different spectral modes are strongly fluctuating and are therefore in the simulation best approximated with a single averaged value.The required averaging due to the intrinsic fluctuations of the FEL pulses washes out the deeper interference valleys.This is evident at longer delays suggesting that the position and appearance of the third trailing pulse is more accidental.The results prove not only the existence of several sub-pulses in the soft xray FEL beam, but for the first time their relative fixed temporal position for a given operation condition of the electron beam.The spatio-temporal coherence | γ12(r, τ) | of the two partial beams is also evaluated for a fixed ∆x = 4 mm at various positions across the fully overlapped profile of both partial beams.Figure 6 shows a fully overlapped interference pattern at zero time delay.It is evident that interference fringes are observed over the whole beam profile.Therefore, at a wavelength of about 24 nm the FEL beam is to a large extend spatially coherent, justifying the use of a wave front division interferometer.The temporal behaviour of | γ12 | is very similar across the beam profile, when particular positions in the horizontal direction of the interference pattern are evaluated.In Fig. 6 a maximal visibility of V = 0.45 is observed.This value drops only to V = 0.39 when positions are analyzed where the electric field at 80% of the radius of one partial beam overlap with that at 20% of the others partial beams radius.This indicates that the transverse coherence is rather uniform over a large part of the beam profile with a diameter of approximately 8 mm, as expected from FEL theory [15,18].It should be kept in mind, however, that an interference pattern with many fringes like that in Fig. 6 also shows a time lapse of up to 6.2 fs across the image.A detailed analysis of the spatio-temporal coherence will thus be presented elsewhere.Previously, the transverse coherence of the vacuum ultraviolet emission of the TTF FEL between 80 and 120 nm had been measured by Ischebeck et al. [10] by a Youngs double slit experiment.At ∆x = 1 mm slit separation a high visibility of up to 0.65 has been observed, which decreases to V = 0.25 for slit separations of ∆x = 2 mm or larger.The value for 1 mm slit separation is in good agreement with our measurements with spatial separation of 0.8 mm.The faster decrease of coherence with increasing spatial separation in the former experiment is mainly caused by the smaller waist of the FEL beam in this experiment.

Conclusion
In summary the soft x-ray radiation typically emitted from a FEL operating under saturated conditions consists of more than a single pulse in time and in frequency, for a single bunch, and observable even when averaged over many pulses.The temporal coherence shows a fast decay with a coherence time of τcoh = 6 fs compatible with the emission of a single Fourier transform limited mode at 23.9 nm.Modulations in the temporal coherence function |γ12| are explained by the emission of independent sub-pulses over an extended period of time.Two main pulses show a separation of about 12 fs.From the interferences also the spatial coherence of the FEL pulses can be assessed.The pulses show a fairly uniform coherence over a large part of the beam profile with very similar time coherence properties.Only at the very edges of the profile a significant reduction of the coherence can be noticed.A significant improvement of the time structure and coherence may be expected for a seeded FEL [21].Alternatively, a spectral sorting of the FEL pulses is required when use shall be made of the full time resolution of a few femtoseconds.

Fig. 1 .
Fig. 1.(a) Schematic drawing of the layout of the autocorrelator.Grazing angles of 3° and 6°f or the fixed and variable delay arms, respectively, are employed to ensure a high reflectivity of the soft x-ray radiation.(b) Calculated reflectivity for amorphous carbon coated silicon mirrors for hν = 30 to 200 eV.The full green line shows the reflectivity of a single mirror for a grazing angle of 6°.The beam size in x-and y-directions are 8 mm, respectively, at the position of the CCD camera.

Fig. 2 .
Fig. 2. Series of interference fringes at 23.9 nm for four different crossing angles of the partial beams from α = 0.18 mrad to 0.75 mrad yielding fringe spacing of δ = 130 µm to 32 µm.

Figure 3 (
Figure 3(a).and 3(b).show single exposure interference fringes at λ = 23.9nm (hν = 51.8eV) at zero delay and at a delay of 55 fs, respectively, when both beams are partially overlapping.The zero delay is derived from the position of the central peak of the symmetric temporal visibility curve, i.e. the normalized temporal correlation function.In addition, the rough zero position has been measured independently using an external broadband visible diode laser.It is evident that interference fringes are clearly visible over the whole overlapped beam area.In the specific case depicted spatial components of the FEL beam separated in the horizontal direction by ∆x = | x1 -x2 | = 800 µm or 20 % of the beam radius are brought to interference.At zero delay an effective visibility of Veff = (0.61 ± 0.04) averaged over the vertical beam profile is observed.From this measurement a degree of coherence of | γ12 | =

Fig. 3 .
Fig. 3. Single exposure interference fringes at 23.9 nm for a crossing angle of α = 0.18 mrad, (a) at zero delay and (b) at 55 fs delay between both partial beams.(c) Observed visibility (experimental data points: red dots) as a function of time delay.The green line depicts a Gaussian function with a coherence time of τ coh = 6 fs, representing a single Fourier transform limited pulse.The red rectangles denote the overlap area of both partial beams.

Fig. 4 .
Fig. 4. Individual single shot spectra of the FEL pulses at about 23.9 nm.

Fig. 5 .
Fig.5.Simulation of the time dependence of the visibility with structured pulses.Separations of pulse maxima of 10 fs (red line), 12 fs (dark blue), and 20 fs (grey) are considered.A third weak maximum with a separation of 40 fs is added to account for an increased visibility at this time delay.Inset: Schematic structure of the best fitting pulse.For this pictorial illustration a flat phase for the individual maxima at a bandwidth of ∆λ = 0.17 nm is chosen.A probably existing chirp of the pulses does not change the time dependence of the visibility.

Table I .
Operating parameters of FLASH during the experiment