Towards ultrafast dynamics with split-pulse X-ray photon correlation spectroscopy at free electron laser sources

One of the important challenges in condensed matter science is to understand ultrafast, atomic-scale fluctuations that dictate dynamic processes in equilibrium and non-equilibrium materials. Here, we report an important step towards reaching that goal by using a state-of-the-art perfect crystal based split-and-delay system, capable of splitting individual X-ray pulses and introducing femtosecond to nanosecond time delays. We show the results of an ultrafast hard X-ray photon correlation spectroscopy experiment at LCLS where split X-ray pulses were used to measure the dynamics of gold nanoparticles suspended in hexane. We show how reliable speckle contrast values can be extracted even from very low intensity free electron laser (FEL) speckle patterns by applying maximum likelihood fitting, thus demonstrating the potential of a split-and-delay approach for dynamics measurements at FEL sources. This will enable the characterization of equilibrium and, importantly also reversible non-equilibrium processes in atomically disordered materials.


Supplementary Figures
Supplementary Figure 1: Analog-to-digital units per droplet collected by the pnCCD detector. Histogram of ADUs per droplet for 2×10 4 split-pulse speckle patterns collected from a suspension of 1-nm-radius Au nanoparticles. The vertical dashed lines refer to the maxima of the photon peaks. The σ value denotes the noise of the detector. The separation between the first and second photon peak is denoted by ∆ P 1,2 .

Supplementary Note 1 : Determination of speckle contrast via Maximum Likelihood Fitting
To extract the contrast β as a function of q from extremely weak speckle patterns, we use the following data analysis steps. Firstly, using a droplet algorithm 3,4 we determine the number of photons detected in each pixel in each splitpulse speckle pattern. Secondly, we determine the split-pulse speckle contrast using a maximum likelihood function approach. These steps are outlined in detail below.

Photon Fitting
An accurate determination of the contrast from very low-count-rate speckle patterns requires that the number of pixels containing one, two, and three photons must be accurately determined for every split-pulse speckle pattern.
A droplet analysis entails identifying adjacent pixels in the area detector where groups of (or individual) photons deposited charge (measured in analog-to-digital units, ADUs). Supplementary Fig. 1 shows a typical histogram obtained from summing the droplets of 2×10 4 sequentially recorded split-pulse frames. In this measurement, the scattering signal from a dilute suspension of 1-nm-radius gold particles in hexane was collected by the pnCCD detector. The histogram peak positions correspond to multiples of about 2500 ADUs. Each droplet is made up of a group of edge-sharing pixels, and contains a sum total of ADUs originating from an integer number of photons.
Thus, the charge from photons that hit the sensor near the edges or corners of pixels is identified and accounted for. Once mapped, each droplet can be used to accurately identify the positions of individual photons by fitting a linear combination of ADU point spread functions to the observed spatial ADU distribution of a droplet. A detailed description of this approach is presented elsewhere 4 . The noise of the detector was derived from the σ value of the first photon peak (208 ADUs) and the separation between the first and the second photon peak ∆ P 1,2 = 2500 ADUs. The resulting histogram shows that the detector can distinguish between 1, 2, 3 photon events with a high sensitivity of 12σ.

Maximum Likelihood Estimation
The contrast of a speckle pattern β is expressed in terms of the number M of coherent modes with 5 β = 1/M.
For a given photon count rate in the detector (λ, photons/pixel) and a number of modes M, the probability of observing i photons in a pixel is given by the negative binomial probability distribution given in the main text by equation (1). This probability distribution is valid for regions of the detector with uniform average scattering. Thus, for each sample and scattering geometry, we divided the detector into regions of interest (ROIs) limited by contours of equivalent q in which the average scattering varies by not more than ∼10 %.
The count rate of each pattern in a given region of interest (ROI) of the detector for a set of speckle patterns (indexed j = 1 . . . J) with a given splitting ratio can be estimated aŝ Here, H is the total number of pixels in the ROI, and N ROI is the number of pixels in the detector ROI. By utilizing this count rate estimate, equation (1) in the main text reduces to a function of a single variable M that can be estimated from n i,j andλ j from a large number of speckle patterns by maximum likelihood fitting 1,2 . A maximum likelihood goodness-of-fit parameter χ 2 ML as a function of M, also known as a log-likelihood ratio statistic, is defined as Minimizing χ 2 ML with respect to M provides an estimate of M with an uncertainty (variance) given by The maximum likelihood approach allows photon statistics from patterns with very low count rates to be meaningfully integrated into the estimate of M of the entire data set. Supplementary Fig. 2  This approach allows for a global evaluation of an entire ensemble of patterns at selected spitting ratio and q value thus minimizing the uncertainty of M and the speckle contrast value. This is an essential advantage compared to the contrast evaluation of individual patterns with the help of negative binomial distribution method 4 .
Supplementary Note 2 : Split-pulse ratio measurements using a single branch monitor FEL sources provide highly transversely coherent ultra short and brilliant photon pulses 6 . Due to the nature of SASE process (i.e., start up from random shot noise) the spectral profile of a single SASE pulse consists of sharp spikes distributed in energy 7 (see Supplementary Fig. 4). Each of these spikes corresponds to a single longitudinal coherent mode 8 . The energy distribution of these spikes changes on a shot to shot basis. Any narrowband spectral filter (such as the narrowband Bragg optics used in this work) will restrict the number of spikes in the transmitted spectrum, leading to more pronounced pulse-to-pulse intensity fluctuations.
The narrowband Si(422) beam splitter of the split-and-delay instrument was set such that two neighbouring, slightly different energies were selected within the bandwidth of the upstream XCS monochromator (∆E = 980 meV) 9 . In this configuration a ∼120 meV portion (given by the Si(422) bandwidth) of the incident pulse was reflected into the upper branch while the remaining portion of the pulse was transmitted through the beamsplitter SP1. The R4 reflector was slightly detuned from the SP1 setting in order to select an energy band that is different from that selected by SP1 and to maximize the intensity of the lower branch. The pulse intensity in the two branches was monitored with two monitor detectors in the upper (UB) and lower branch (LB), respectively (see Fig. 1 in the main article). Since the two monitors see different parts of the spectrum the correlation between the monitors is weak (see Supplementary Fig. 3) and not suited to determine the splitting ratio (e.g., by a simple division of the pulse intensities).
Instead we used the strong correlation between the signal of the lower branch (I LB ) and the sum of split-pulse intensity from both branches at the pnCCD (I pnCCD ). As described above, the energy acceptance (bandpass) of a pair of Si(422) crystals (SP1 -R1 or R4 -R5) is about ∆E = 120 meV. Thus about 3 spikes from a typical FEL SASE pulse ( Supplementary Fig. 4a red line) are transmitted through (SP1 -R1) to the upper branch monitor or through (R4 -R5) to the lower branch monitor. However, the acceptance function is Gaussian rather than box shaped, yielding a transmission function (see the red line in Supplementary Fig. 4b) that leads to unity transmission for spikes with energy E 0 but reduced transmission for spikes corresponding to the tails of the bandpass (E = E 0 ± ∆E).
After the monitor, the pulse is transmitted through another pair of Si(422) crystals (R2 -R3 or R6 -SP2) with another Gaussian energy transmission function that further restricts the total bandpass, shown in Supplementary   Fig. 4b (blue). As a result, correlations of pulse intensity measurements made after the first pair of crystals (I LB ) with pulse intensity measurements after all four crystals (I pnCCD ) will not be linear because different bandpasses apply. This nonlinearity can be understood by considering two cases, and we restrict our discussion to comparing the lower branch monitor (I LB ) with the total counts in the pnCCD (I pnCCD ), as this applies to our work. First, we consider a pulse with energy near E 0 that passes the first pair of crystals with minimal intensity loss and registers a high I LB value. Assuming perfect optical alignment, this pulse will continue through the the second pair of crystals, again without loosing much intensity, because E 0 is identical for both pairs of (422) crystals and the energy width of a single FEL spike is narrower than the 4-crystal bandpass of the split-and-delay. On the other hand, pulses that do not contain a spike at E 0 will have relatively low intensity at I LB because the bandpass will attenuate all spikes with E 0 ± ∆E. When such pulses pass through the second pair of crystals, the total intensity I pnCCD will be even lower because the 4-crystal energy filter further attenuates the FEL spikes with energies in the tails of the Gaussian bandpass.
We thus expect underproportional pnCCD signals for I LB monitor count rates. However with increasing I LB monitor high signal transmission through all four crystals is much more likely, leading to a linear correlation regime of I pnCCD with I LB at higher monitor values. This non-linear behaviour was modeled by using a simple 3 spike model with equal intensity probabilities and computing the intensity for 1000 three spike events after propagation through one (red) or two (blue) filter functions (Si(422) crystal pairs). The result is shown in Supplementary Fig. 4c showing the expected non-linear correlation which can be described phenomenologically by a simple error function (dashed line in Supplementary Fig. 4c). In order to cast the functional behaviour of the F LB = 1 in a single functional form we defined a function f D via where the parameter R is a rotation matrix calculated for the angle of 40 • . The error function and R were chosen such that f D describes a slightly curved line (dashed line in Fig. 3a in the main text). The F LB = 0 boundary corresponds to the horizontal (mean pnCCD count rate) I pnCCD axis in the plot. Based on the functional form of the F LB = 1 profile and according to one can establish contours on the correlation diagrams that delineate populations of split-pulse speckle patterns with equivalent F LB (the fraction of intensity passing through the lower branch, see the F LB bands the figure).
The contrast of a split-pulse correlation function depends on the splitting ratio (or F LB ) as shown in Fig. 3c in the main text. Analyzing all fractions together would average out the contrast. The reliability of the estimate of M obtained via maximum likelihood depends on the integrated intensity for a F LB band. It is thus necessary to identify the split-pulse frames for the most intense fraction (here F LB = 0.625). Supplementary Fig. 5 shows a histogram of the number of frames, the integrated intensity and the mean count rate I (photons/pixel/frame) as a function of F LB . The histogram peaks (as expected) at F LB = 0.5 indicating the most probable splitting ratio is 1:1 but it does not distinguish between intense and weak equally split pulses. The width of the distribution shows the distribution of splitting ratios seen in the correlation diagram. The integrated intensity diagram is shown in Supplementary   Fig. 5b) and is identical to the Fig. 3b in the main text. From the intensity distribution( Supplementary Fig. 5a) and the probability distribution one can derive the mean intensity I per frame ( Supplementary Fig. 5c) which peaks at 0.85. This result is in fact expected since the lower branch receives on average more intensity.

Supplementary Note 3: Split-pulse speckle contrast
The experimentally observed contrast of the optical setup depends on i) the dynamics of the sample, ii) the initial contrast, iii) the pulse splitting ratio r sp and iv) the degree of decohence σ d between the two split beams. The influence of the aforementioned parameters is discussed here.
Correlation times of the investigated system are measured by analyzing split-pulse speckle patterns at a selected time delay ∆ t as a function of the wave vector transfer q. Supplementary Fig. 6a shows the contrast variation calculated using equation (2) given in the main text with D 0 = 7.3 × 10 −10 m 2 ·s −1 and ∆ t = 1.3 ns and different values of β 0 (i.e, the contrast at q = 0). For a fully transversely coherent beam, β 0 equals one. For large q values, when the dynamics of the sample is faster than ∆ t , the contrast is reduced to half of its initial value, i.e.
lim q→∞ β(q, ∆ t ) = β 0 /2. Any incoherent contribution of the incident beam to the recorded split-pulse speckle patterns will lead to a decrease of β 0 which affects the contrast function β(q, ∆ t ), as shown in Supplementary Fig. 6a .
The static contrastβ 0 measured with a single branch of the split-and-delay line at q = 0.03 nm −1 is 0.66 ± 0.04.
This value was obtained under experimental conditions where the measured speckle size was much larger than the pixel size of the detector (S s > P s ). Under the current experimental conditions the expected speckle size S s was 45 µm. This size was estimated from the horizontal and vertical beam size where λ s is the wavelength and L is the sample to detector distance, respectively. The decrease of the static contrast due to detector resolution (pixel size P s ) was estimated 10 via β 0 =β 0 (1 + (P s /S s ) 2 ) −0.5 and gives a value of β 0 = 0.32 ± 0.05. Supplementary Fig. 6b shows the expected contrast function described by equation (2) in the main text for β 0 = 1 and various splitting ratios r sp . The strongest reduction of speckle contrast occurs for r sp = 1. Other splitting ratios will reduce the amplitude of contrast variation.
The split-pulse speckle contrast function β is also affected by other factors that introduce decoherence between the two beams including geometrical overlap and differences in incident angles and wavelength between the split beams illuminating the sample. In an ideal experiment, the only source of decoherence measured in the splitpulse speckle patterns would stem from the dynamics of the sample. However, under real experimental conditions one can expect a deviation from such conditions. The validity of equation (3) given in the main text has been verified for the limiting cases of σ d (i.e., σ d = 1 and σ d = 0) by simulating the overlap between the two split pulses and calculating its effect on the split-pulse speckle contrast. The simulations involved 4336 spherical particles undergoing Brownian motion placed on a grid of 2000×2000 lattice points and illuminated by a beam of about 7825 pixels as shown in Supplementary Fig. 7a. The contrast of the incident beam β SB was set to 1. The speckle pattern generated for the perfect overlap between the two equal intensity pulses, r sp = 1 is shown in Supplementary   Fig. 7b. Supplementary Fig. 7d shows the speckle pattern for the zero overlap case. The intensities corresponding to a single q band were selected (region between the black rings in Supplementary Fig. 7b) and the contrast β 0 was calculated according to β 0 = Var(I)/ I 2 . For σ d = 1 ( i.e., the two split pulses fully overlap ) the speckles are clearly visible (see the inset of Supplementary Fig. 7b and the contrast is β 0 = 1. In case of zero overlap (σ d = 0) the speckles are smeared out (as shown in the inset of Supplementary Fig. 7d and the resulting β 0 equals to 0.5, as predicted by equation (3) given in the main text.

Supplementary Note 4 : Correlation times of gold nanoparticles following Brownian motion
The relaxation times τ c of a monodisperse sample undergoing Brownian motion are calculated according to 11 τ c (q) = (D 0 q 2 ) −1 (6) where D 0 is the free particle diffusion coefficient (see Methods section in the main text). Supplementary Fig. 8 shows the expected relaxation times calculated with the help of supplementary equation (6) for the wave vector range corresponding to measured scattering angles. The time τ c at q < 0.4 nm −1 is longer than 10 ns. In this range the sample behaves mostly as if it were static when probed with pulses separated by 1.3 ns. Accordingly, the contrast measurements in this q range approach the expected value for a static sample β = 0.29 ± 0.02, indicated by the upper dashed line in Fig. 4 in the main text. At q > 1 nm −1 , the dynamic time constant τ c is about 1 ns or shorter. Under these conditions, the split-pulse speckle pattern will be decorrelated and the contrast will decrease.
Complete decorrelation will result in a contrast of 0.17 ± 0.02 (lower dot-dashed line in Fig. 4 in the main text).