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We report on the feasibility of high wavevector temporal speckle correlation measurements at the world’s first hard x-ray free electron laser, the Linac Coherent Light Source (LCLS). Due to the chaotic nature of LCLS, the spectral profile of the x-ray radiation fluctuates on a pulse-to-pulse basis. Its impact on the determination of the single shot speckle contrast in a wide angle x-ray scattering geometry is investigated by analyzing FEL power spectra that are simulated based on the nominal operational parameters of LCLS. Ultimately, a potential scheme to deliver a single-mode hard x-ray pulse is proposed by using a narrow bandpass crystal monochromator.
© 2012 Optical Society of America
1. Introduction
The principle of a self amplified spontaneous emission (SASE) free electron laser (FEL) has been demonstrated at various wavelengths [1–5]. Since then, much excitement has been generated by the perspective to perform coherent x-ray scattering experiments at 4th generation light sources that are expected to deliver unprecedented peak brilliance, femtosecond pulse duration and a laser-like degree of coherence in the hard x-ray regime. The Linac Coherent Light Source is the first of the hard-x-ray FELs in operation and serving users since October 2009 [6] followed by SACLA [7].
One of the pioneering attributes of 3rd generation light sources was their capability of producing partially coherent x-ray beams several orders of magnitude more intense than previously available [8]. The access to coherent x-rays has enabled experiments such as x-ray photon correlation spectroscopy (XPCS) [9] and coherent diffraction imaging [10]. XPCS in particular, the analogous of dynamic light scattering in the x-ray regime, relies on the characterization of time-resolved speckle patterns [8, 11, 12]. A unique approach, named split and delay [15], offers the possibility to access ultrafast time scales (fs∼ns) in a wide variety of systems of interests such as amorphous materials. In this method, a single x-ray pulse is split into two pulses by a silicon crystal beam splitter [13, 14]. Two pulses are systematically delayed with respect to each other by a controlled time delay and then recombined before being directed to the sample. Finally the scattered radiation from the sum of the two pulses is collected by a 2D imaging detector. Given a sufficient photon flux, the speckle contrast, i.e. the contrast of coherent x-ray diffraction patterns, measured by the detector in a coherent x-ray diffraction experiment can be evaluated via where σ(S(Δt)) and S(Δt) denote the standard deviation and the intensity on the detector as the function of the time delay Δt [15]. If the characteristic correlation time of the dynamics in the sample is longer than the time delay between two pulses, the contrast from the summed up image is equal to that from a single image. On the other hand, when the delay time becomes greater than the correlation time, the speckle patterns from two delayed beams begin to differ, which results in a reduced contrast of the summed up image. Thus, XPCS at FEL sources is capable of providing insight in the dynamics of complex disordered system on length and time scales inaccessible by other techniques [12, 15, 16]. However the successful implementation of such a technique requires the detailed observation of speckle patterns that need to be measured with sufficient contrast. The speckle contrast is intimately linked to the coherence properties of the incident beam and has been limited by the partial coherence available from storage ring based light sources.
An FEL carries different operational performances as compared to high stability storagering-based 3rd generation sources because of the SASE process itself. It provides x-rays with full transverse coherence. However due to the start-up random shot noise in the electron beam [17, 18], a single SASE FEL pulse consists of a large number of coherent wave-packets, which results in many spikes appearing in the output spectrum [19, 20]. These spectral spikes correspond to longitudinal modes. The LCLS beam thus behaving as a multi-mode laser but with mode fluctuation on a shot-to-shot basis. Since the contrast of a speckle pattern depends on both the degree of coherence of the beam and the optical configuration of the experiment, a comprehensive understanding of the coherence properties of FEL sources is of a fundamental interest and a necessity in order to take full benefit of this new light source.
We present a numerical investigation of the shot-to-shot fluctuations of the speckle contrast expected at the Linac Coherent Light Source. The input parameters for the calculation are modeled from the currently available LCLS operational parameters [6]. We also propose a scheme that can provide enhanced capabilities, i.e. single-mode laser-like x-ray pulse, by using different x-ray monochromators. Our result is consistent with the recent result from a wide angle coherent x-ray scattering experiment [21].
2. Coherence properties of hard x-ray beams from 3rd and 4th generation light sources
The coherence properties of 3rd generation synchrotron sources are described by the wavelength spread of the photons Δλ/λ and the phase space volume (ΣΣ′)2 in which the photons are contained, where Σ is the source size and Σ′ the divergence of the electron beam. Partial coherence requires ΣxΣ′xΣyΣ′y ≤ (λ/4π)2, which is usually not fulfilled for wavelengths in the hard x-ray regime. The coherent portion of the beam can be selected using apertures which reduces the coherent flux of photons per 0.1% bandwidth Ic = (λ/2)2 · B, where B is the brilliance of the source given in units of photon/s/mm2/mrad2/0.1 %. Today’s 3rd generation sources can provide a brilliance of the order of 1020 [8].
The coherence properties of such sources are quantified by their transverse and longitudinal coherence lengths. The transverse coherence length is ξt = (λ/2)(R/s), where R is the distance from the electron source and s the source size. It is of the order of 10 to 100 μm at third generation sources for λ =1 Å (cf. Ref. [8]). The longitudinal coherence length ξl = λ2/Δλ is typically 100 Å for an undulator with Δλ/λ ≈ 1 % at λ =1 Å. It can be increased to several microns by using a monochromator, which narrows the bandpass of the x-ray beam (cf. Table 1). The increase of the coherence length, however, comes at the expense of the coherent flux available to perform an experiment. Fourth generation light sources intrinsically provide orders of magnitude more brilliant x-rays (B > 1033) due to the nature of the SASE process. The transverse coherence length of FEL sources is larger or equal to the x-ray beam size, and thus exhibits full transverse coherence [22]. The beam is said to be diffraction limited. However, the longitudinal coherence properties fluctuate on a shot-to-shot basis, which will translate into speckle contrast fluctuations. The details of these fluctuations must be addressed in order to correctly interpret coherent x-ray scattering experiments.
Table 1. Monochromaticity Δλ/λ, the theoretical longitudinal coherence length ξl and mean speckle contrast, mean number of expected x-ray photons and its fluctuation from 700 single shot spectrums. All numbers summarized in the table assume that LCLS first harmonic is at λ = 1.384 Å.
3. Analysis scheme
3.1. Simulation of the LCLS FEL beam spectra
The spectral structure of the LCLS x-ray beam is generated using 1-D FEL simulation near the saturation point [19]. The input parameters for our numerical study uses the operational parameters of LCLS. The solid black line in Fig. 1 shows an example of a single-shot spectrum of the first harmonic at λ = 1.381 Å (i.e. E=8.97keV) for 250 pC electron bunch charge and 100 fs (FWHM) electron bunch duration. The averaged spectrum from 1000 shots is described by a smooth Gaussian distribution with a FWHM corresponding to Δλ/λ ≈ 0.1 % centered at λ ≈ 1.384 Å as indicated by the gray solid line. The bandwidth of the x-ray radiation can be further reduced by inserting a crystal monochromator, which improves the longitudinal coherence properties. Figure 1(Inset) also shows theoretical rocking curves of three different silicon Bragg reflections, overlayed on top of a single FEL spectrum for comparison. Under these operational parameters, we note that an individual spike has an average width of Δλ/λ ≈ 4.9·10−6 that is narrower than the bandwidth of the three silicon reflections. The presence of fluctuations of the longitudinal coherence length leads to speckle contrast fluctuations on a pulse-to-pulse basis [21]. Our study investigates the use of a monochromator as a possible mitigation for these fluctuations and its impact on coherent x-ray scattering experiments.
Fig. 1 The black line displays a single shot LCLS spectrum simulated with the LCLS operation parameter (250 pC bunch charge with the first harmonic at λ = 1.381 Å). The solid gray lines is the average of 1000 single-shot spectrums and presents a smooth Gaussian distribution. The Full Width at Half Maximum gives Δλ/λ ≈ 0.1 % for each bunch charge. Inset shows the comparison between the single shot spectrum and three possible monochromator settings to be used at LCLS where blue square, red triangle and green circle data points correspond to diffraction curves of Si (111),(220) and (511) reflections respectively.
The effect of monochromatization is calculated using the DuMond approach [23], in which the Bragg reflected intensity is determined from the overlap between the source divergence and the intrinsic bandwidth of the monochromator single crystal at a particular wavelength. Figure 2(Left) displays a single-shot Bragg reflected spectrum for three different silicon crystals. One can clearly observe the effect of each crystal that reflects a smaller number of spikes as the respective wavelength acceptance reduces. For instance, while ≈16 temporal modes are transmitted through Si(111) on average, only 1 to 2 modes remain after Si(511). The single-shot spectrum after the monochromator is not described by a smooth Gaussian envelope, but depends strongly on the detailed intensity distribution of the spikes contained in it. Since it is non-trivial to define a bandwidth Δλ/λ or equivalently the longitudinal coherence length ξl, there is no simple analytical way to evaluate the speckle contrast.
Fig. 2 Left : Bragg reflected intensity from a single-shot spectrum after various silicon monochromator crystals: (111), (220) and (511). Right : Corresponding normalized mutual coherence function obtained from the Fourier Transform of each spectrum presented on the left, as described in Sec. 3.2
3.2. Evaluation of the single-shot speckle contrast
The speckle contrast from x-ray scattering as a function of wave vector transfer and sample geometry is investigated by following the formulation derived by Pusey [24], which has been effectively used in previous studies [25–27]. It is shown that the contrast of a speckle pattern is given by
where Γ(r⃗,t) is the mutual coherence function (MCF) at position r⃗ and time t = Q⃗ · (r⃗2 − r⃗1)/cko where Q⃗ is the scattering vector, ko = 2π/λ is the wave number, c is the speed of light, E is electric field, and V is the illuminated sample volume. Here, since the x-ray beam at FEL facilities such as LCLS has been measured to be transversely fully coherent [21], Eq. (1) becomes a volume integral of the MCF over V, which depends on the experimental geometry. The MCF can be directly calculated from the Fourier Transform of the power spectral density according to Wiener-Khinchin theorem. The resulting MCF consists of the fine details of single shot spectrum whose FWHM corresponds to the longitudinal coherence time. Due to its complex structure, an evaluation of the speckle contrast β requires a numerical integration.In order to quantify the effect of the spectral fluctuation on the resulting speckle contrast, we perform the simulation by modeling the experimental parameters that were used in a recent wide angle scattering experiment at LCLS [21]. As for the input parameters, we used a x-ray spot size of 9(h) x 3(v) μm2 and a sample thickness of 2.24 μm. We modeled an area detector which pixel size is 20 x 20 μm2 placed 1.55 m downstream the sample to collect scattered x-rays at a scattering angle of 34° corresponding to Q = 26 nm−1. This wavevector corresponds to a time delay of about 5 fs.
Figure 2(right) shows the normalized MCF computed from the respective band-passed spectrum from Fig. 2(left). The shaded area under the solid blue lines represents the fraction of the mutual coherence function that is involved in the volume integration of Eq. (1). We note that throughout the rest of our work, we are primarily interested in the time delay 5 fs corresponding to the maximum path length difference as indicated by the red dotted line. We expect that, for the monochromaticity of Si (111) and Si (220), the speckle contrast is more sensitive to the intricate structure of the coherence function. On the other hand, nearly perfect longitudinal coherence should be expected from Si (511) where the coherence length easily exceeds the path length difference of the scattered x-rays. In order to perform a statistical study on the pulse-to-pulse fluctuations of the speckle contrast, we generated a few hundred spectrums and evaluated them using Eq. (1).
Due to the SASE nature of LCLS, the beam spectral profile strongly fluctuates, which results in fluctuations of the MCF. As an illustration, the red line in Fig. 3 shows MCF of high coherence cases where a single spectrum after the respective monochromator contains a few dominant temporal modes closely centered at a particular wavelength. On the other hand, the dotted blue lines shows a typical low coherence cases where the maximum number of possible temporal modes are evenly distributed within the bandwidth of the monochromator. For Si (111) and Si (220), the differences between two extreme cases of the MCF are very pronounced. The profiles of MCF changes dramatically depending on the intricate structures of the spectrum. For Si (111), the low coherence MCF (dotted blue lines) shows a coherence time of 2 fs while in the high coherence case, it represents a considerably longer coherence time of about 5 fs. Similar trends are observed for Si (220). However, the difference is marginal for Si (511), which effectively transforms the spectrums with multiple temporal modes into a single mode x-ray pulse under the LCLS operational parameters used in this study. This is an interesting feature as the Si (511) monochromator could enable the delivery of a single-mode hard x-ray laser beam.
Fig. 3 Profile of mutual coherence function respect to different settings of monochromator. Solid red shows high coherence case while dotted blue shows low coherence case.
In order to compensate for the reduction of coherent flux associated with a monochromator rocking curve narrower than the single mode width, it is proposed that the mode width should be optimized for each monochromator crystal. The width of a single spike in an energy spectrum corresponds to the x-ray pulse width that is controlled by varying the electron bunch width, Tb. Tb is given by the relation, electron bunch charge/peak current. Ultimately the x-ray pulse width can be adjusted by changing the electron bunch charge. For example, LCLS operates with the bunch charge of 250 pC and a peak current of 3000 A and thus generates x-rays with a pulse duration of 100 fs. On the other hand, a bunch charge of 20 pC at the same peak current results in a considerably shorter x-ray pulse width of 10 fs. We estimate that the optimized bunch charge should be 80 and 20 pC for Si (511) and (220) respectively. In order to use Si (111) in a single-mode scheme, the bunch charge would be unfortunately too low for the proper operation of LCLS. Operating LCLS with a “custom” bunch charge offers the possibility to match the rocking curve of monochromators crystals and thus to perform coherent scattering experiments with a single-mode-like hard x-ray laser. This also optimizes the available coherent flux for a given monochromator setting. This indeed is an attractive scheme until the hard x-ray seeding capabilities become available [28, 29].
4. Analysis on the speckle contrast fluctuation and method of compensation
The successful interpretation of XPCS experiments at LCLS relies on the understanding of the statistical reliability of the contrast and its associated uncertainty. For most transmission geometry (i.e. Small Angle X-ray Scattering), the relevant time difference is sufficiently less than 1 fs. In this case, Si (111) should be be used because it provides a sufficient longitudinal coherence to fulfill the condition of coherent sample illumination and also provides the highest coherent photon flux (See Table 1). However, a considerably higher longitudinal coherence is required for experiments measuring coherent scattering patterns at large angles [25–27] where most of the ultrafast dynamics are expected to be observed.
Figure 4 shows series of speckle contrast evaluated from the simulated FEL spectrums for three different monochromator settings. After 700 iterations, we observe that the average speckle contrast under the experimental geometry converges to 0.259, 0.422 and 0.568 at Q = 26 nm−1 for the monochromators Si (111), (220) and (511) respectively. For the case of Si (111), our numerical result is consistent with the experimental data of Gutt et al [21]. We note that the experimental data shows slightly more pronounced speckle contrast fluctuations. This implies that there were less temporal modes transmitted by the Si (111) monochromator. Therefore, it would indicate that the x-ray pulse duration is shorter than the electron bunch width at LCLS, of which detail is explained elsewhere [21, 30].
Fig. 4 Speckle contrast results for three different monochromator settings of Si (111), (220) and (511) for a 250 pC bunch charge.
5. Effect of speckle contrast fluctuation on XPCS measurements
The spectral fluctuation in the FEL beam translates into fluctuations of speckle contrast at high wave vectors because the path length differences between the scattered x-rays from the sample are comparable to the longitudinal coherence length of the x-ray. Such jitter in the speckle contrast will generate noise in a split-delay correlation measurement. Without sufficient statistics, the noise level would potentially overwhelm the signal from sample. Therefore, it is highly desirable to anticipate how many x-ray pulses need to be accumulated to provide reliable measurement.
In order to investigate this effect, we simulated the 2D Brownian random motion of particles, from which individual snap shots of scattering images are generated via Fourier Transform of the real image. The correlation dynamics of this system is calculated via the split-delay approach [15]. Figure 5 shows the time delay dependence of the contrast C(Δt) at Q = 26 nm−1 for various shot averaging. We assume that the particles remain stationary during the time of x-ray illumination. For each monochromator setting Si(511), (220) and (111), the correlation function decays from maximum values of 0.568, 0.422 and 0.259 respectively. Significant noise contributions are observed when Si (111) and (220) are used. The effect from the noise are especially pronounced when 1 or 10 single shot speckle patterns are used to evaluate the contrast. The level of the fluctuation due to the noise nearly overwhelms the decaying behavior of the speckle contrast. The contrast evaluated from 100 speckle patterns resolves the decaying behavior of the Brownian sample dynamics. However, for Si (511), the fluctuation becomes considerably smaller up to a point where even a single image can used to determine distinctive decorrelation behavior. This indicates that Si (511) would be most appropriate for performing wide angle XPCS experiments at LCLS.
Fig. 5 Speckle contrast correlation function C(Δt) for the monochromator settings of Si (111), (220) and (511) and for different number of x-ray pulses cumulated. Red square, blue star and green circles represent 1, 10 and 100 shots averaged respectively.
High monochromaticity comes at the considerable expense of x-ray photon flux. The LCLS pink beam carries intrinsic fluctuations in its central wavelength (0.1 % RMS) and flux (5 to 12 % RMS) [6]. A monochromator selects the wavelength within a given Δλ/λ (cf. Table 1) and thus converts the pink beam central wavelength fluctuations into intensity fluctuations. An estimate of the intensity variation is presented for each configuration in Table 1. These shot-to-shot variations clearly increase when reducing Δλ/λ. Although the intensity fluctuation can almost reach 100 % in selected cases, it can be accounted for in the data analysis by monitoring the incident flux. Fortunately, with the current repetition rate of LCLS (120 Hz), such level of averaging can be obtained in a matter of seconds.
We also note that a certain degree of temporal broadening is expected after the monochromators due to the crystal diffraction geometry and uncertainty relation [31]. Nevertheless marginal broadening of 2 to 5 fs should be insignificant under the nominal operation condition of LCLS. For the low charge operational mode (e.g. 20 pC) where sub 10 fs x-ray pulse duration can be achieved, the effect of the temporal broadening needs to be taken into account, which is beyond the scopes of our study.
6. Summary
We present the result of a numerical simulation on the pulse-to-pulse fluctuations of the speckle contrast of LCLS in the hard x-ray regime of large wavevector. The effects of various monochromators, as required to perform coherent x-ray scattering experiments such as XPCS was investigated in detail. These fluctuations can be remedied with a sufficient pulse-averaging within a reasonable data acquisition time. We also propose a scheme to generate a near single-mode x-ray laser beam by using a Si (511) monochromator and by tuning electron bunch charge. This provides an attractive option to perform coherent x-ray scattering experiments at large wave vectors until the availability of a seeded hard x-ray free electron laser [28, 29].
Acknowledgments
The Linac Coherent Light Source is funded by the U.S. Department of Energy’s Office of Basic Energy Sciences and led by the SLAC National Accelerator Laboratory, which is operated by Stanford University for the DoE. The authors would like to thank David Fritz and Eric Landahl for proofreading the manuscript.
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26. A. R. Sandy, L. B. Lurio, S. G. J. Mochrie, A. Malik, G. B. Stephenson, J. F. Pelletier, and M. Sutton, “Design and characterization of an undulator beamline optimized for small-angle coherent x-ray scattering at the Advanced Photon Source,” J. Synchrotron Rad. 6, 1174–1184 (1999). [CrossRef]
27. O. K. C. Tsui, S. G. J. Mochrie, and L. E. Berman, “Statistical analysis of x-ray speckle at the NSLS,” J. Synchrotron Rad. 5, 30–36 (1998). [CrossRef]
28. Y. Ding, A. Brachmann, F.-J. Decker, D. Dowell, P. Emma, J. Frisch, S. Gilevich, G. Hays, Ph. Hering, Z. Huang, R. Iverson, H. Loos, A. Miahnahri, H.-D. Nuhn, D. Ratner, J. Turner, J. Welch, W. White, and J. Wu, “Measurements and simulations of ultralow emittance and ultrashort electron beams in the Linac Coherent Light Source,” Phys. Rev. Lett. 102, 254801 (2009). [CrossRef] [PubMed]
29. G. Geloni, V. Kocharyan, and E. Saldin, “Cost-effective way to enhance the capabilities of the LCLS baseline”, http://arxiv.org/pdf/1008.3036.pdf.
30. S. Lee, Linac Coherent Light Source, SLAC National Accelerator Laboratory, 2575 Sand Hill rd, Menlo Park CA 94025, and A. Robert are preparing a manuscript to be called “Single shot characterization of the coherence properties of the Linac Coherent Light Source by means of speckle visibility.”
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