Realizing split-pulse x-ray photon correlation spectroscopy to measure ultrafast dynamics in complex matter
Yanwen Sun, Mike Dunne, Paul Fuoss, Taito Osaka, Aymeric Robert, Mark Sutton, Makina Yabashi, Diling Zhu
RRealizing split-pulse x-ray photon correlation spectroscopy to measure ultrafastdynamics in complex matter
Yanwen Sun, ∗ Mike Dunne, Paul Fuoss, Aymeric Robert, and Diling Zhu † Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California, 94025, USA
Taito Osaka and Makina Yabashi
RIKEN SPring-8 Center, Sayo, Hyogo, 679-5148, Japan
Mark Sutton ‡ Physics Department, McGill University, Montr´eal, Quebec, H3A 2T8, Canada (Dated: January 15, 2020)Split-pulse x-ray photon correlation spectroscopy has been proposed as one of the unique capa-bilities made possible with the x-ray free electron lasers. It enables characterization of atomic scalestructural dynamics that dictates the macroscopic properties of various disordered material systems.Central to the experimental concept are x-ray optics that are capable of splitting individual coherentfemtosecond x-ray pulse into two distinct pulses, introduce an adjustable time delay between them,and then recombine the two pulses at the sample position such that they generate two coherentscattering patterns in rapid succession. Recent developments in such optics showed that, while true‘amplitude splitting’ optics at hard x-ray wavelengths remains a technical challenge, wavefront andwavelength splitting are both feasible, able to deliver two micron sized focused beams to the samplewith sufficient relative stability. Here, we however show that the conventional approach to specklevisibility spectroscopy using these beam splitting techniques can be problematic, even leading to adecoupling of speckle visibility and material dynamics. In response, we discuss the details of theexperimental approaches and data analysis protocols for addressing issues caused by subtle beamdissimilarities for both wavefront and wavelength splitting setups. We also show that in some scat-tering geometries, the Q -space mismatch can be resolved by using two beams of slightly differentincidence angle and slightly different wavelengths at the same time. Instead of measuring the visibil-ity of weak speckle patterns, the time correlation in sample structure is encoded in the ‘side band’ ofthe spatial autocorrelation of the summed speckle patterns, and can be retrieved straightforwardlyfrom the experimental data. We demonstrate this with a numerical simulation. I. INTRODUCTION
Nearly fully transversely coherent femtosecond x-raypulses produced by x-ray free electron laser (FEL)sources opened up the possibilities of direct measure-ment of atomic scale dynamics of complex systems attheir native time scales [1]. One area of particular in-terest is the investigation of noncrystalline matter suchas liquids, glasses, amorphous and disordered systems,and holds the promises of unlocking the mysteries be-hind the glass transition, liquid-liquid phase transitions,fragile-to-strong transitions, to name a few [2–4]. A pri-mary methodology with the potential to extend dynamiclight scattering to angstrom and femto-/picosecond timescale is the so-called split-pulse x-ray photon correla-tion spectroscopy (XPCS) technique, where the dynam-ics of the scattering object are imprinted onto the fluc-tuations of coherent scattering intensity distribution [5].The schematic of a generic split-pulse XPCS experiment ∗ Also at Physics Department, Stanford University, Stanford, Cal-ifornia, 94305, USA † [email protected] ‡ Also at Linac Coherent Light Source, SLAC National AcceleratorLaboratory, Menlo Park, California, 94025, USA is illustrated in Fig. 1. Two delayed beams are gener-ated by a split-delay optics, and then focused down to asmall size at the sample location. Downstream the sam-ple a pixelated X-ray detector measures coherent scat-tering patterns. While area detectors capable of inde-pendently measuring the scattering patterns from twosubsequent x-ray pulses with a femto- to picosecond sep-aration will not be available in the foreseeable future, itwas proposed that the correlations between the coher-ent scattering patterns from the two successive pulsescan nevertheless be obtained from the summed scatter-ing pattern, by analysing the speckle visibility [6]. Thedependence of the visibility, as a function of of the tem-poral separation between the two pulses, thus carries thepotential to provide detailed information on the dynam-ics information of the system being probed [7].The purpose of x-ray split-delay lines is to generatex-ray pulse pairs with continuously adjustable time sep-arations in the femto- and picosecond time range. Thegeneric split-delay-recombine optical arrangement hasbeen realized recently with increased robustness, primar-ily in the form of wavefront or wavelength splitting se-tups. Pulse pairs can now routinely be generated anddelivered to a sample with sufficient reliability and sta-bility for two pulse coherent scattering measurements [8–10]. In this work, we present detailed examinations of a r X i v : . [ phy s i c s . i n s - d e t ] J a n FIG. 1. Schematics of a generic split-pulse XPCS experimentusing crystal optics based hard x-ray split-delay. A set ofcrystals are arranged such that individual pulses are split,delayed in time as compared to each other and subsequentlyfocused and recombined at the sample location. Arrows alongthe green beam path indicate how this path length can beadjusted by moving some of the crystals within the split-delayoptics. The summed coherent scattering patterns measuredfor each pulse pair is recorded by an x-ray imaging detectorlocated downstream the sample at a given scattering angle. the speckle correlation analysis in these scenarios, illus-trate the incompatibility of the wavefront and wavelengthsplitting optical schemes with the speckle visibility spec-troscopy concept. We propose an alternative correlationextraction methodology, as well as a Q -space compen-sation solution by using two different wavelengths, thatallows the extraction of dynamics under the general ex-perimental scheme of two-pulse XPCS. We also discussoptimization of real experiment parameters. II. SPLIT-PULSE SCATTERING GEOMETRY
The two-pulse XPCS measurement concept envisionedthe use of two identical x-ray pulses, i.e, having the samephoton energy, trajectory, beam profile, wavefront, andcoherence properties, with an adjustable time separation.This was initially proposed to be realized by using thincrystal optics with thickness smaller than the extinctiondepth of the chosen x-ray Bragg reflection [11]. However,the fabrication and handling of sufficiently thin and ro-bust beam splitting crystals still remains to date a majortechnical challenge.Two alternative splitting techniques, wavelength andwavefront splitting, have been adopted during the pastfew years in Bragg crystal based x-ray split-delay op-tics [8, 10–13]. While these systems have shown greatprogress towards delivering two similar x-ray foci tothe sample with fine control of their time delay andgood relative beam position stability, these splitting tech-niques lead to other ineluctable differences in the twobeams/pulses. For example, Roseker et al. and Osaka etal. used thin silicon crystals as beam splitters [11, 12].However, the available thin crystals are still thicker thanthe extinction depth of the reflection. As a result, theportion of wavelengths that falls within the reflectingbandwidth gets almost fully reflected, while the otherwavelengths transmit through the crystal. The two out- put beams as a result will have different photon ener-gies. More recent x-ray split-delay optics adopted thewavefront splitting geometry [8, 10, 13]: part of the in-coming beam hits a polished edge of the beam splittingcrystal, meets Bragg condition and gets reflected, whilethe other part of the beam passes over the edge. Thesplit beams are directed into different beam paths withinthe split-delay optics before getting recombined using an-other crystal with a polished edge. In this case, the twoparts of the recombined beam are parallel but not exactlycollinear. Experimentally, when trying to bring the bothparts of the beam to the same location on the samplewith focusing optics, there will be an inevitable crossingangle between the two beams.The slight differences in the two ‘probe’ beam proper-ties will lead to a mismatch in their scattering in the farfield, which could in principle compromise our ability torecover the desired material dynamics. Below we providea generalization of this mismatch originating from thosedifferences.
FIG. 2. Illustration of the Ewald spheres considering the dif-ferences between the two recombined beams.(a) Coordinatesystem definition (b) Illustration of the mismatch of scat-tering vectors in the reciprocal space. The two beams aredenoted in orange and green. They can either have slight dif-ferent wavelengths corresponding to the radius change of theEwald sphere or different incident angles corresponding to therotation of the Ewald sphere.
As shown in Fig. 2(a), we define z as the incident beampropagation direction and y as the direction along whichtheir trajectories deviate from one another. Using theexit beam wavevector k f , we can define the sphericalcoordinate: 2 θ as the angle with respect to z axis, φ as the angle of its projection on xy plane (BM) and x axis (0 ≤ φ < π ). The scattering experiment can bepresented as shown in Fig. 2(b) in the reciprocal space:with two Ewald spheres denoted in orange and green forthe two output beams. Their two slightly different radius k i and k (cid:48) i represent the difference between their photonenergies. Its length is thus related to the difference inthe wavelength δλ : δk i = k (cid:48) i − k i = k (cid:48) f − k f = δλλ k i . (1) k i = AO and k (cid:48) i = A (cid:48) O are the incidence wavevectors. η is the angle between the two indicating their slightdifferent incident angle on sample. We use k f = AB and k (cid:48) f = A (cid:48) B (cid:48) for the two output wavevectors. A cho-sen detector pixel can be represented by the parallel exitwavevectors k f (cid:107) k (cid:48) f for the two beams respectively. Inthe Cartesian coordinate system defined by x, y, z , theincidence and exit wavevectors for both beams can bewritten as: k i = AO = k i [0 , , , k (cid:48) i = A (cid:48) O = k (cid:48) i [0 , sin η, cos η ] , k f = k i [sin 2 θ cos φ, sin 2 θ sin φ, cos 2 θ ] , k (cid:48) f = k (cid:48) i [sin 2 θ cos φ, sin 2 θ sin φ, cos 2 θ ] . (2)The difference in the momentum transfer at the samedetector pixel location BB (cid:48) = OB (cid:48) − OB can be derivedas: BB (cid:48) = ( k (cid:48) f − k (cid:48) i ) − ( k f − k i )= ( k (cid:48) i − k i )[sin 2 θ cos φ, sin 2 θ sin φ, cos 2 θ ]+ [0 , − k (cid:48) i sin η, k i − k (cid:48) i cos η ] . (3)An area detector samples the speckles that lie on the twoEwald spheres separately for the two beams. BB (cid:48) is ameasure of the deviation of the momentum transfer Q measured by the same detector pixel.We next discuss this Q mismatch for wavefront andwavelength splitting schemes respectively and the re-sulting constraints on the experimental geometry andsample parameters. For the rest of the paper, we willchoose a photon energy of 10 keV, and a bandwidth δλ/λ = 5 . × − (FWHM) and momentum transferof interest at Q = 2 ˚A − corresponding to θ ≈ . ◦ forexperimental case studies. A few assumptions are madefor speckle size calculation and we follow the methodsexplained in details in Ref. [14–16]. III. WAVEFRONT SPLITTING CASE
In this section we discuss the case of wavefront split-ting. Figure 3(a) is a schematic of the realization ofthe split-delay based on polished edge crystals. Afterbeam recombination at the crystal beam combiner, thetwo output beams travel nearly collinearly in order toachieve spatial overlap at the sample using focusing op-tics, as illustrated in Fig. 3(c). The magnitude of theminimum crossing angle η is therefore determined by thebeam width w (defined as in Fig. 3(c)) of the unfocusedbeam and the focal length f : η ≈ wf . (4) FIG. 3. (a) Optical arrangement of a split-delay system basedon wavefront splitting using crystals with polished edges. (b)illustration of the wavefront splitting/combining process asindicated by the dashed oval in (a). (c) Illustration of thecrossing angle between the two beams after focusing opticsdue to the non collinear geometry.
We also assume the two output beams have the samephoton energy, and thus k i = k (cid:48) i . The Q space mismatchthen reduces to BB (cid:48) = [0 , − k i sin η, k i (1 − cos η )]. Itsmagnitude BB (cid:48) = 2 k i sin( η/
2) is invariant of θ and φ . BB (cid:48) can be decomposed into its in - and out-of-detector-plane components. In order for the same pixel to bemapped to the same speckle, the magnitude of the out-of-detector-plane mismatch BC should be much smallerthan the speckle ellipsoid size along the exit wavevectordirection ( k f or k (cid:48) f ). Otherwise the detector will be sam-pling a completely different slice of the 3D Q space. Thetwo speckle patterns will have no correlation as a result.The out-of-detector-plane mismatch can be written asBC = BB (cid:48) · k f k f = k i [ − sin 2 θ sin φ sin η + cos 2 θ (1 − cos η )]= − k i η sin 2 θ sin φ + k i O ( η ) . (5)Here we denote the sum of all higher order terms of η as O ( η ) because typically η is on the order of 10 − consid-ering the small numerical aperture of the x-ray focusingoptics. It has a dependence on both 2 θ and φ . To firstorder, the magnitude of BC is maximum for φ = π/ φ = 0 where it is k i O ( η ).Similarly, the in-detector-plane mismatch: CB (cid:48) = BB (cid:48) − BC = k i [ 12 η sin θ sin 2 φ + O ( η ) , ( − θ sin φ ) η + O ( η ) , η sin 4 θ sin φ + O ( η )] . Its lengthCB (cid:48) = k i η (cid:113) − sin θ sin φ + O ( η ) (6)reaches a maximum of k i η for φ = 0 and a mini-mum of k i η cos 2 θ for φ = π/
2. The direction of in-detector-plane mismatch is s = [0 , ,
0] for φ = 0 and s = [0 , − cos 2 θ, sin 2 θ ] for φ = π/ FIG. 4. Illustration of wavefront splitting for φ = π/
2. (a) Re-ciprocal space illustration of the scattering of two pulses fromsplit-delay optics using wavefront splitting and recombining.The two pulses are plotted in orange and green and have acrossing angle η between the incident wavevector k i = AO and k (cid:48) i = A’O . The incident beam bandwidth is indicated by thethickness of the Ewald circle in shade orange/green. At thesame detector location (2 θ with respect to k i ), the measuredwavevectors are respectively OB and OB’ . (b) A zoomed inview of the rectangular area in (a). The blue arrow representsthe momentum transfer mismatch BB (cid:48) . A speckle ellipsoidis plotted in light gray and the detector locations are plottedin black for the two Ewald circles. The speckle size, on the other hand, is determined tothe first order by the focal spot size and the sample thick-ness t . The largest possible speckle size is reached at thediffraction limited focal spot size of w : w ≈ π λw f. (7)Following Ref. [17], the rms speckle size in the y direc-tion S y ≈ . k i λ/w ≈ . k i η . Define c as the ratiobetween the rms speckle size S y and the scattering mis-match BB (cid:48) ≈ k i η , i.e., c = αS y k i η , (8)where α = √ φ = 0, along s , the rms speckle size S s = S y , while the in-detector-planemismatch is k i η . c ≈ .
73 suggests that the Q spacemismatch in the detector plane is generally larger thanone speckle size. This leads to the unavoidable reductionin contrast in the sum-speckle pattern. As a result, vis-ibility analysis will become significantly less sensitive tosample dynamics.For φ = π/ s , the rmsspeckle size is S s = S y S z (cid:113) S z cos θ + S y sin θ , (9)where S z is the rms speckle size along z , which is to firstorder determined by the thickness of the sample [18] S z ∼ k i λt . (Note that longitudinal coherence also plays a role. Forcomputing S z , we follow the numerical methods providedin Ref. [14].) Its ratio with respect to CB’ is αS s CB (cid:48) ≈ c cos 2 θ (cid:113) cos θ + S y S z sin θ< c cos θ ≈ . . (10)Just as the φ = 0 case, here the in-detector-plane mis-match is inevitably larger than the speckle size. We willprovide an analytical solution to address the in-detector-plane mismatch in later sections. However, the out-of-plane mismatch will have to be minimized. Ideally, BD,the out-of-detector-plane speckle size defined in Fig 4(b),shall be much larger than the out-of-plane Q mismatch,i.e., 2BDBC (cid:29) . (11)For our case study, at φ = π/
2, followingBD = αS y S z (cid:113) S z sin θ + S y cos θ , we have 2BDBC = c sin 2 θ (cid:113) sin θ + S y S z cos θ< c sin θ ≈ . , (12)when S y << S z , or when the sample thickness is muchsmaller than w . Typical values of w = 100 µm , f =1 meter give η = 10 − , BC ≈ . × − ˚A − , w ≈ . µm and S y ≈ . × − ˚A − . Experiments require to usevery thin samples, t ∼ nm , which limits the totalscattering signal.For φ = 0, considering a sample thickness of t = 15 µm , S z ≈ . × − ˚A − , BC = k i O ( η ) ∼ − ˚A − , we have2BDBC ≈ . × (cid:29) . Clearly, in order to minimize the out-of-plane Q mis-match, the φ = 0 configuration would be more advan-tageous than φ = π/
2. However, the in-detector-plane Q mismatch would still make speckle visibility spectroscopyinfeasible. Alternative correlation extraction method willbe discussed in a later section. IV. WAVELENGTH SPLITTING CASE
FIG. 5. Optical arrangement of a split-delay system usingthin crystal wavelength splitting.
We now evaluate the wavelength splitting scenario.The schematics of this type of split-delay optics is illus-trated in Fig. 5. Similar to the wavefront splitting case,wavelength splitting also leads to a mismatch in Q spacesampling between the two beams, even though the twobeams can be recombined with high degree of collinear-ity. This is because the magnitude of k i and k (cid:48) i will beslightly different as a result of the wavelength difference.This is illustrated in Fig 6. The Q space mismatch BB (cid:48) can be written as: BB (cid:48) = δk i (sin 2 θ cos φ, sin 2 θ sin φ, − θ ) . (13)Using Eq. 1, its length BB (cid:48) = δλ/λQ . Here Q = 2 k i sin θ is the momentum transfer for the k i (orange) and then Q (cid:48) = 2 k (cid:48) i sin θ would be for the k (cid:48) i (green). Similarly,we can derive the the ‘in’ and ‘out’ of detector planemismatch: CB (cid:48) ∼ δλλ Q cos θ (14)BC ∼ δλλ Q sin θ (15) FIG. 6. Illustration of wavelength splitting. (a) Reciprocalspace illustration of the scattering of two pulses from split-delay optics using wavelength splitting and recombining. OA and OA’ are incident beam wavevectors which have differ-ent magnitudes ∆ k i = AA (cid:48) . At the same detector locationof scattering angle 2 θ , the measured wavevectors are respec-tively OB and OB’ . Using this method, the two pulses havedifferent center energies offset by the bandwidth of the crystalreflection. (b) A zoomed in view of the rectangular area in (a).A speckle ellipsoid is plotted in light gray and the detectorlocations are plotted in black for the two Ewald circles.
Both in- and out-of-detector-plane mismatches have nodependence on φ . Assuming S x = S y in this case,the out-of-detector-plane speckle size BD and the in-detector-plane speckle size S s are also only related toscattering angle 2 θ . The ratios between the speckle sizesand the magnitude of the Q mismatch are therefore: αS s CB (cid:48) = αS y S z Q cos θ (cid:113) S z cos θ + S y sin θ · λδλ , αS y S z Q sin θ (cid:113) S z sin θ + S y cos θ · λδλ . With the same chosen experiment parameters providedearlier, BC ≈ . × − ˚A − and CB (cid:48) ≈ . × − ˚A − for δλ/λ = 5 . × − . For a 15 µm thick sample, BD ≈ . × − ˚A − and S s ≈ . × − ˚A − . The ratios ofspeckle size and the mismatch in and out of the detectorplane are thus still of comparable magnitude, calculatedto be αS s / CB (cid:48) ≈ . / BC ≈ . V. Q -SPACE COMPENSATION ANDCORRELATION EXTRACTIONA. Compensation of the out-of-detector-planemismatch Following the formalism presented in the previous sec-tion, the out-of-detector-plane mismatch caused by thecrossing angle between the two beams can be fully com-pensated by an intentional wavelength mismatch, as il-lustrated in Fig. 7(a). One could expand the green Ewald
FIG. 7. (a) Illustration of combing wavefront and wave-length splitting to miniize the out-of-detector-plane compo-nent of the Q mismatch BB’ . (b) A zoomed in view of thedeviation between the Ewald spheres at Q denoted by OG af-ter crossing at Q denoted by OB , we have ∠ GAB = δ (2 θ ) asthe angle covered before the out-of-detector-plane mismatchis too large. sphere around O such that the two Ewald sphere crosseach other again near B and B’. The goal is to have δ Q = Q − Q (cid:48) , or the vector BB’ , in the tangential direc-tion of the Ewald sphere. In this configuration, within asmall Q region near point B , the detector samples close-to-identical slices in the reciprocal space. In other words, BB’ is perpendicular to AB , so in the triangular BOB’,we have OB (cid:48) sin(180 ◦ − θ ) = OBsin( θ − η/ . And this gives us ∆ λλ ∼ η tan θ . (16)The same relationship can be obtained by equating theright side of Eq. 5 and Eq. 15. At 10 keV, use Q = 2˚A − as the momentum transfer of interest, with η = 10 − ,we derive the required difference of the wavelength of the two split-delay branches to be ∆ λ/λ ≈ . × − , whichis well within the SASE pulse bandwidth [22].Another quantity we need to estimate is the scatteringangle coverage δθ , which is how large in scattering anglethis method can correct before the out-of-detector-planemismatch of momentum transfer becomes non-negligible,i.e., at point G and G’ in Fig. 7(b). The deviation in theout-of-detector-plane direction is δ BC ≈ δ ( ∆ λλ Q sin θ ) − δ ( ηk i sin 2 θ )= 2 k i ηδθ. (17)Using the parameters mentioned above, BD ≈ . × − ˚A − . As η = 10 − , δ BC = BD / ≈ . × − ˚A − means δ (2 θ ) ≈ .
029 (or ∼
144 mm at 5 meter de-tector distance). This can be translated to covering N = 2 k i δ (2 θ ) /S s ≈ . × speckles before the out-of-detector-plane mismatch increases to of significant influ-ence (1/2 of BD, the out-of-detector-plane speckle size). B. Treatment for in-detector-plane mismatch
As shown in Fig. 7, even though the out-of-detector-plane Q mismatch is well compensated by using both dif-ferent wavelength and incident angles, the in-plane mis-match cannot be cancelled, we have the in-plane mis-match: BB (cid:48) = OB sin( η/ θ − η/ ≈ k i η + k i O ( η ) (18)The speckle patterns from the two branches will havean offset in the direction of crossing. As η ∼ − , theoffset is to the first order invariant of scattering angle2 θ . For the beam parameters discussed above, BB (cid:48) ≈ . × − ˚A − is larger than in-detector-plane specklesize, and the sum of the speckle patterns will be shiftedby tens of speckle sizes. As a result, the visibility analysiswhich calculates intensity correlation from the scatteringof the two branches at the same detector location willnot work.In this case, the dynamics information regarding thesample can be extracted via the spatial intensity auto-correlation of the summed speckle patterns. Using i, j toindicate the pixel p i,j falling into the chosen ROI on a2D detector, and assuming there is a vertical mismatch s in the speckle pattern between the two pulses. s corre-sponds to the BB’ in the reciprocal space as mentionedabove. f denotes the frame number recorded. Using ∆ t to denote the time separation between the two pulses ina pulse pair, define I f = I ,f ( t ) + I ,f ( t + ∆ t ) , the intensity correlation between pixel p i,j and p i,j + s canbe estimated with the following equation: A ( p i,j , s, ∆ t ) = 1 N f (cid:80) N f f =1 ( I ,f ( p i,j , t ) + I ,f ( p i,j , t + ∆ t ))( I ,f ( p i,j + s , t ) + I ,f ( p i,j + s , t + ∆ t ))( I ( p i,j ) + I ( p i,j ))( I ( p i,j + s ) + I ( p i,j + s )) (19)Intensity average for each pixel p i,j is I n ( p i,j ) = 1 N f N f (cid:88) f =1 I n,f ( p i,j ) (20)Here n = 1 , r as the fraction of the first pulse intensity: r = I I + I (21) A ( p i,j , s, ∆ t ) = I ( p i,j ) I ( p i,j + s ) I ( p i,j ) · I ( p i,j + s )= 1 + r − r + I ( p i,j ) I ( p i,j + s ) I ( p i,j ) · I ( p i,j + s ) (22)Averaging over ROI covering an iso- Q range, we have: A ( Q, s, ∆ t ) = 1 N ROI (cid:88) i,j ∈ ROI ( Q ) A ( p i,j , s, ∆ t ) . (23)Here N ROI is the number of pixels enclosed in the ROI.Siegert relation [23] states that g ( Q, ∆ t ) = (cid:104) I I s (cid:105)(cid:104) I (cid:105)(cid:104) I s (cid:105) = 1 N ROI (cid:88) i,j ∈ ROI ( Q ) I ( p i,j ) I ( p i,j + s ) I ( p i,j ) · I ( p i,j + s )= 1 + β | f ( Q, ∆ t ) | . (24)Here I s indicates the speckle pattern of the second pulseshifted by s in order to get aligned with that of the firstpulse. The spatial intensity correlation encodes sampleinformation: A ( Q, s, ∆ t ) = 1 + r − r + r (1 − r ) g ( Q, ∆ t )= 1 + ( r − r ) β | f ( Q, ∆ t ) | . (25)The only additional assumption is that correlationsshould show negligible variation over δQ = BB’. When r = 0 .
5, this equation describes the equal intensity case: A ( Q, s, ∆ t ) = 1 + 14 β | f ( Q, ∆ t ) | . (26)In conclusion, in the detector plane, as scattered photonsfrom Q and Q + δQ fall into the same location on thedetector, we need to measure coincidence of photons δQ apart. And this can be calculated directly via the spatialcorrelation of the recorded 2D scattering sum, with thedecorrelation between the two speckle patterns revealedin the decrease of the side band peak magnitude. VI. SIMULATION OF THE SOLUTION
FIG. 8. Simulation of speckle patterns using wavelength andwavefront splitting assuming the two x-ray beams illuminat-ing a 1 . µm × . µm × µm volume of random scatter-ers. The detector is placed at vertical scattering geometry12.4 meter from the sample with 50 µm pixel size. (a) Thespeckle pattern of the beam in orange with center photon en-ergy 10 keV. Its lower left corner corresponds to 2 θ ≈ . ◦ and φ = 90 ◦ with respect to the beam denoted in orange asshown in Fig. 4 and 6. (b) The speckle pattern of the beam ingreen as illustrated in Fig. 6 with a difference in center wave-length δλ compared to the orange beam ( δλ/λ = 5 . × − ).(c) The speckle pattern of the green beam as illustrated inFig. 4 with η = 10 − in the vertical direction. Both beamshave the same center photon energy. (d) The speckle patternafter using a different photon energy for the green beam as il-lustrated in Fig. 7 to compensate for the out-of-detector-planemismatch of scattering caused by the crossing angle η = 10 − .Here the difference in center photon energy or wavelength ∆ λ satisfies ∆ λ/λ ≈ × − . The white dashed boxes in (a) (c)(d) enclose the same 2 θ and φ range. Using the same beam parameters, we performed a sim-ulation by calculating the coherent scattering from anillumination volume of 1 . µm × . µm × µm ran-dom scatterers, with 15 µm being the sample thicknessalong the beam direction k i . A detector with 50 µm pixel size was placed 12.4 meter downstream the samplein the vertical scattering geometry ( φ = 90 ◦ ) to over-sample the speckles in the scattering. Shown in Fig. 8(a)is the scattering of the nominal beam denoted in orange FIG. 9. Spatial correlations of the simulated speckle pat-terns in Fig 8 in the vertical direction. (a) Orange: in-tensity autocorrelation of the nominal speckle pattern asshown in Fig. 8(a). Purple: intensity cross correlation be-tween the nominal and the other beam from wavelength split-ting (Fig. 8(b)). Gray: intensity cross correlation betweenthe nominal and the other beam from wavefront splitting(Fig. 8(c)). Green: intensity cross correlation between thenominal and the compensated speckles (Fig. 8(d)). (b) Or-ange: intensity autocorrelation of the nominal speckle patternas shown in Fig 8(a). Purple: intensity autocorrelation of thespeckle sum of the nominal and the other beam from wave-length splitting (Fig. 8(b)). Gray: intensity autocorrelationof the speckle sum of the nominal and the other beam fromwavefront splitting (Fig. 8(c)). Green: intensity autocorrela-tion of the sum of the nominal and the compensated specklepatterns (Fig. 8 (d)). as shown in Fig. 4, 6 and 7 with 10 keV center pho-ton energy and beam incidence along the sample thick-ness direction. The lower left corner of the speckle pat-tern corresponds to a momentum transfer of Q = 2˚A − (2 θ = 22 . ◦ ) and φ = 90 ◦ . Due to the vertical scatter-ing at high angles and the illumination dimension nearlyan order of magnitude larger along the incident beamdirection, the speckle size is smaller in the vertical di- rection on the detector. Its intensity autocorrelation inthe vertical direction is plotted in orange in Fig. 9(a)and (b) as reference. The small side lobes are due to thenon-Gaussian illumination. We plotted in Fig. 7(b) and(c) the speckle patterns of the green beams illustratedin Fig. 6 and Fig. 4 that slightly deviate in the centerwavelength or incident angle from the orange beam dueto the wavelength/wavefront splitting. Using wavelengthsplitting, the out-of-detector-plane mismatch is a factorof 2.6 smaller as compared to the speckle size. The in-plane mismatch component leads to the speckles shiftingin the vertical direction by almost one speckle size. Asdisplayed in Fig. 7(b), we can still visualize shifted butsimilar speckles with a change in the intensity distribu-tion. The center peak shift and value reduction in thecross correlation between this speckle pattern and thenominal one plotted in purple in Fig. 9(a) confirm bothin- and out-of-plane mismatch from our previous calcu-lation. This is neither optimized for visibility nor spa-tial intensity correlation analysis when only their specklesum can be measured. The autocorrelation of the sumis also drawn in purple in Fig. 9(b). From this we cansee that the shift leads to a broader center peak andcontrast reduction to close to 0.6. It is important tonote here that it is impractical to make the differenceof the wavelengths of the two pulses ∆ λ larger in or-der to fully separate the same speckle measured by thetwo beams, as this will require extremely small samplethickness due to the increase of the out-of-detector-planemismatch, which is also proportional to ∆ λ/λ . Narrowbandwidth reflection ∼ − will be preferred for opti-mizing the geometry for visibility analysis.For Fig. 7(c), even though k (cid:48) i (green) has a crossingangle η with respect to that of k i , we still choose the 2 θ to be the scattering angle of the exit wavevector with re-spected to k i as this relates to the same location on thedetector. We can see that as the out-of-detector-planespeckle size is very small compared to the speckle mis-match with 2BD / BC ≈ .
30 in this case. The detectoris actually sampling different speckle ellipsoids. As a re-sult, we are not able to identify similar speckle patternsany more. Its cross correlation with the nominal specklepattern together with the autocorrelation of their sumplotted in gray in Fig. 9(a) and (b) suggest that the de-tector cannot detect correlation anymore as it is imag-ing different speckles in the reciprocal space. Shown inFig. 7(d) is the speckle patterns after we use a differentcenter photon energy of the green beam to compensatefor the effect of the crossing angle. As mentioned, with∆ λ/λ ≈ . × − , the out-of-detector-plane mismatchcan be fully compensated, this is why we can again vi-sualize the exact same speckles, as indicated by the pinkdashed box. However the in-detector-plane mismatch iseven larger as the effects from the crossing angle anddifferent wavelengths add up. The information regard-ing sample dynamics can be extracted from the shiftedspeckle sum using spatial intensity correlation analysis asmentioned in the previous section. The cross correlationof the nominal and compensated speckle patterns plot-ted in Fig. 9(a) shows that the two speckle patterns areshifted but highly correlated, and the autocorrelation oftheir sum is plotted in green in Fig. 9(b) where we cansee two side lobes with correlation value equal to 1.25,which is what we calculated using r = 0 . VII. DISCUSSIONA. Mitigation with long beamline
FIG. 10. A general source-to-sample schematics including thesplit-delay optics and the focusing optics.
So far we have only discussed about the scenario as-suming the split-delay optics is much closer to the focus-ing optics compared to the distance to the source. Thisis the case for most current systems being deployed atthe x-ray FEL facilities. In this case, the angular specklesize and the crossing angle will be always on the sameorder. There is a possibility to reduce the crossing angleif space allows for a very long beamline and installingthe split-delay optics far upstream closer to the source.We now consider a more general split-delay optics instru-ment layout as shown in Fig. 10, with a goal of reducingthe crossing angle while still maintaining a high level ofbeam overlap at the sample location. We assume the or-ange beam is on the optical axis of the lens. One canrotate the last crystal to steer the green beam path bya small angle δ to make the two beams achieve partialspatial overlap at the lens. This effectively introducesan offset vertically to the source of the green beam withrespect to the original (orange) source position by theamount d = L SD δ. (27)In the lens imaging system with a focal length f , assume f is on the order of a few meters, and L is on the orderof a few hundred meters, then the distance between thelens and the demagnified source image f (cid:48) ≈ f . The shiftof the green beam focus can be estimated by d (cid:48) = d f (cid:48) L ≈ L SD L δf. (28)In order to have the focus shift much smaller than thefocus size, i.e., w >> d (cid:48) . Using Eq. 7, enforcing that d (cid:48) is 10 times smaller than w , we obtain the relation of δ ≈ . λw LL SD . (29)Here we notice that the factor L SD /L , with the split-delay closer to the source, demagnifies the virtual sourceshift.On the other hand, following the earlier discussion aswell as the schematics shown in Fig. 3, the crossing angleafter the focusing lens can be now written as η ≈ w + d (cid:48) − ( L − L SD ) δf = [ L SD L − L − L SD f ] δ + wf . (30)This presents an opportunity to minimize η by choosing L and L SD to fulfill the relationship of: wf ≈ [ L − L SD f − L SD L ] δ. (31)If we use the typical values of λ ≈ − and w ≈ L × − , we will arrive at L SD = LL/
13 + f /L + 1 <
13 meter , or the split-delay system must be unrealistically close tothe source to fulfill such requirement. One could workaround this potentially by working with a beam size w that has been slit down. For example, if we slit down thebeam by a factor of 4, such that w ≈ L × − /
4, we willarrive at L SD ≈
137 meter which is more realistic, at acost of reduced photon flux.
B. High-speed signal processing with photoncoincidence measurements
In Section V B, we proposed a spatial correlation anal-ysis scheme for handling the momentum transfer mis-match in the detector plane. This bears similarity to arelated concept in dynamic light scattering introduced forsuppressing multiple scattering known as the 3D cross-correlation light scattering. The concept utilizes a sym-metric detection setup, where the information regardingdynamics at a momentum transfer Q can be studied viathe cross correlation of the signal measured separatelyat Q and − Q [24, 25]. We also note that using twopulses of slight different wavelengths to compensate forthe out-of-detector-plane momentum transfer mismatchis very similar to the two-color dynamic light scatteringexperiments demonstrated in the 1990s [26]: By usingtwo lasers with different colors at a crossing angle cor-responding to their wavelength difference, it is possibleto also suppress multiple scattering while retrieving thetemporal fluctuations in the scattering. As the same mo-mentum transfer is located at two different spatial loca-tions for the two colors and the detection can be color0filtered, sample information is thus also encoded in thecross correlation of the signal measured. For the aboveDLS experiments, thanks to the extremely high coher-ent flux of optical lasers, fast point detectors measuringthe correlations of a speckle pair are sufficient to achieveenough signal-to-noise ratio.For XPCS studies of atomic scale dynamics using x-rays at FEL sources, scattering signal is typically sig-nificantly less than 0.1 photons per speckle per detectordata acquisition window [18]. The low count rate can bemitigated by the use of large area 2D pixel array detec-tors for simultaneously measurement of as many speck-les in the scattering as possible. In our spatial correla-tion analysis, ‘correlation’ signal comes from the pairs ofspeckles at a distance δQ defined by the crossing angle ofthe two beams. In the case where detectors cannot tem-porally distinguish the two scattering patterns and thusonly record the sum of the two patterns, the observablebecomes the rate of coincidence of photons in the scatter-ing sum separated by δQ . Instead of retrieving specklevisibility by looking at photon counting statistics, thecoincidence rate can be relatively easily extracted froma 2D sensor array by employing a field programmablegate array based spatial corrector on board the x-ray de-tector [27]. This alleviates significantly the burden ofreading out and storing the full image data. In face ofthe upcoming increase of the source repetition rate andmulti-mega-pixels detectors, this provides an effective av-enue towards taking full advantage of the various newtechnologies, and can render ultrafast XPCS using x-rayFEL sources an effective probe of the dynamics in com-plex matters. VIII. CONCLUSION
In summary, we presented detailed analysis of the Q space sampling in the context of split-pulse XPCS experimental concept and the current split-delay opticsimplementations. We provide also discussions of thepractical impact based on real experimental parametersat existing x-ray FEL beamlines. We show that theout-of-detector-plane momentum-transfer mismatch ofthe scatterings needs first to be reduced to well belowthe speckle size along that direction in order to preservethe correlation between the two successive scatteringpatterns from the pulse pair. For the in-detector-planespeckle mismatch, which renders visibility spectroscopyinfeasible, we show that dynamics can still be extractedfrom the summed speckle patterns by spatial intensityautocorrelation analysis. We propose a method usingtwo pulses of different photon energies to compensatefor their different incident angles in the case whenbeam crossing angle is in the scattering plane. Thesemodification to the data collection and analysis protocolare critical for realizing two-pulse XPCS for the mea-surement of ultrafast equilibrium dynamics in complexmatter. ACKNOWLEDGMENTS
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