Pulse-to-pulse wavefront sensing at free-electron lasers using ptychography

1 Pulse-to-pulse w a v efron t sensing at free-electron lasers using pt yc hograph y Simone Sala, a,b 1 Benedikt J. D aurer, c,d Michal Odst r cil, e Fla vio Capotondi, f Emanuele Pederso li, f Max F. Hantke , g Michele Manfred d a , f N. Duane Loh, d,g Pierre Thibaul t b * and Filipe R. N. C. Maia c * a Dep artment of Physics & Astr onomy , U niversity Col le ge L ondon, L ondon , UK , b Dep artment of P hysics & Astr onomy, University of Southamp ton, Southampt on, UK , c Dep artment of Cel l and M ole cular Biolo gy, U ppsala University, Uppsala, Swe den , d Dep artment of Biolo gic al Scienc es, N ational University of Singap or e, Singap or e , e Paul Scherr er Institut, Vil ligen, Switzerland , f Elettr a-Sinc r otr one T rieste, T rieste, Italy , g Dep artment of Chemistry, Oxfor d Unive rsity, O xfor d, UK , and h Dep artment of Physics, National University of Singap or e, Singap or e . E-mail: pierr e.thib ault@soton.ac.uk , filip e@ xr ay.bmc.uu.se Abstract The pressing need for the detailed wa vefron t prop erties of ultra-bright and ultra-short pulses pro duced by free-electron lasers (FELs) has sp urred the deve lopment of sev eral 1 Current address: MAX IV Lab oratory , Lund Universit y , Lund, Sw eden PREPRINT: Journal of Applie d Crystal lo gr aphy A Journal of the International Union of Crystal lo gr aphy 2 complemen tary c haracterizat ion app roac hes. He r e we pr esen t a m ethod based on pt ychog- raphy that can r etriev e full high-resolution complex-v alued w a ve fu nctions of individu al pulses. Our tec hniqu e is demonstrated within exp erimental conditions su ited for d iffrac- tion exp eriments in their nativ e imaging s tate. This lensless tec hniqu e, applicable to many other sh ort-pulse instrum ents, can ac hiev e diffraction-limited r esolution. 1. Introduction F ree-e lectron lasers (FELs) are op ening the wa y to a num b er of new researc h paths. Within the field of microscop y , the highly coherent and short pulses pro duced b y FELs are us ed to conduct diffr activ e imaging of ind ividual particles, also called flash X-ra y imaging (FXI), p oten tially do wn to atomic resolution (Neutze et al. , 2000; Ch apman et al. , 2006; Seib ert et al. , 20 11). Many other in v estigations exploit FEL tigh t focal sp ots to maximiz e flu ence or impro ve spatial reso lu tion (Willems et al. , 2017; Vidal et al. , 2017 ; Min cigrucci et al. , 2018). F or all these applications, a reliable h igh-resolution c haracterization of the sh ot-to -sh ot fo cal sp ot is crucial. A num b er of b eam diagnostics metho ds ha ve b een pur p osely designed for this task. How ev er, with ultrash ort pu lses (down to fem toseconds) w hic h can reac h a flux sufficient to destro y or irr eversibly damage most targets, these metho d s mostly provide only partial information ab out the wa vefron t, suc h as p osition, size, shap e, in tensity or curv ature (Chalupsky et al. , 2011; V artan y ants et al. , 2011; Rutishaus er et al. , 2012 ; Loh et al. , 2013; Sik ors k i et al. , 2015; Keitel et al. , 2016 ; Daurer et al. , 2017). A recen t grating- based metho d can provide real-time wa v efront d istributions (Sc hn eider et al. , 2018; Liu et al. , 2018 ), though a wa y fr om the fo cal plane and with a resolution limited by the grating’s man uf acturing pro cess. IUCr macros version 2.1.11: 2019/01/ 14 3 A t third -generatio n sync hr otron sources, pt yc h ograph y is n o w a p opular w a vefron t char- acterizat ion to ol (Kewish et al. , 2010; Schropp et al. , 2010; T ak ahash i et al. , 2011; H¨ onig et al. , 2011; Vila-Coma m ala et al. , 201 1), thanks to its abilit y to retrieve the complex-v alued w av efield in or close to the fo cal plane (the prob e) along with th e transmission function of the sample (the ob ject) (Th ibault et al. , 2009; Maiden & Ro den bu rg, 2009). Recent impro vemen ts to ptyc h ographic reconstru ction algorithms add ress additional s ources of data degradation, suc h as partial coherence (Thibault & Menzel, 2013 ), scanning p osition jitter (Guizar-Sicairos & Fienup, 2008; Maiden et al. , 2012; Bec kers et al. , 2013; Zhang et al. , 2013; T ripathi et al. , 2014) and prob e v ariations (Odstrcil et al. , 2016). T h e reco v- ered illumination wa v efields can b e numerically p ropagated to refine the fo cal p osition or to reve al optics-induced ab errations. Here w e show that ptyc hograph y can b e u sed to r eliably reconstruct the w a vefron t of individual FEL pulses. Our approac h requires no a priori information on either ob ject or prob es, and all in d ividual wa v efron ts are allo w ed to v ary , in cluding their p ositions. Unlik e a previous ap p licatio n of pt yc hography at the FEL w hic h fi ltered the pu lses through additional optics (Sc hropp et al. , 2013), our method retriev es w a vefron ts in th e nativ e imaging state, i.e. the same conditions in wh ic h the apparatus is used for imaging or diffraction exp eriments. 2. Metho ds The exp eriment was carried out at the DiProI in strument of FERMI (see Ap p endix for details), a seeded FEL p ro ducing 10 µ J pulses, at a p hoton energy of 83 eV , equiv alen t to a w av elength of 15 nm. Figure 1 giv es a sc hematic representa tion of the exp erimen tal setup. IUCr macros version 2.1.11: 2019/01/ 14 4 After atten uation by ab out 4 orders of magnitud e to a vo id damage, the b eam was f o cu sed b y a pair of p erp endicular b endable Kir k p atric k-Baez (KB) mirrors to a fo cal sp ot size of under 10 × 10 µ m 2 . Th e sample – p ositioned close the fo cal plane – w as a gold test p attern featuring a 30 × 30 µ m 2 Siemens star whose S E M image is represented in Fig. 2a. At 83 eV the sample b eha ve s as a b inary ob ject since th e gold-plated areas absorb completely the inciden t X-ra ys . F ar-field diffraction patterns w ere detected with a CCD camera lo cated 150 m m do wnstream from th e sample. KB mirr ors diffraction pattern sample (object) slits Fig. 1. Diagram of the exp erimenta l setup. Adju s table slits form a p u pil that admits the cen tral part of the FEL b eam. Kirkpatric k-Baez mirr ors fo cus the b eam on to a small area of the sample, wh ic h is scanned with a translation stage in the x - y plane. The in tensity of the resulting free-space propagated exit wa v es (i.e. the diffraction patterns) are recorded b y a detector d o wnstream along z . IUCr macros version 2.1.11: 2019/01/ 14 5 a b 10 µm Fig. 2. SEM image (a) and amplitude of ptyc hographic reconstruction (b) of a Siemens star test p attern; the color bar in (b) r epresen ts tr ansmission b etw een 0 and 1. Pt yc h ographic data were colle cted by scanning the sample o ver the Siemens star area. A wide jitter of the p hoton b eam p osition, observ ed thr oughout d ata acquisition and caused b y vibr ations of an upstream optical comp onen t, was first corrected using a cross-correlatio n approac h and then further refined. T his pro cedur e, which would normally not b e n eeded in a routine application of th e metho d , r educed th e num b er of v alid frames down from 1515 to 937. The pt yc hographic reconstruction w as carried out using a single-pulse retriev al algorithm deriv ed from the orthogonal pr ob e relaxation p t yc hograph ic (OPRP) metho d (O d strcil et al. , 2016) whic h reco v ers a differen t prob e for eac h diffraction p attern while k eeping the problem o v er-constrained through dimensionalit y redu ction. Ind ivid ual prob es are thus mo deled as linear com binations of a num b er of d ominan t comp onen ts (also called “mo des” or “eigenprob es”) reco ve r ed d ynamically and without a priori information. As usual for pt ychog r aph y , the ob ject’s transmission f unction is also retriev ed w ith ou t enforcing any prior knowledge . The essence of OPRP is compatible with any p t yc hographic reconstru c- IUCr macros version 2.1.11: 2019/01/ 14 6 tion algorithm and has b een implemented within the PtyPy reconstruction suite (Enders & Thibault, 2016) for b oth difference map (DM) (Thibault e t al. , 2009) and maxim um lik eliho o d (ML) (Thibault & Guizar-Sicairos, 2012) algo rith m s. Th e reconstructions pr e- sen ted here were obtained usin g DM follo we d by ML r efinemen t, b oth using a 10-comp onent decomp osition of the retriev ed pr ob es. 3. Results The absolute v alue of the retriev ed ob ject tr ansmission fun ction is repr esen ted in Fig. 2b, whic h can b e compared with the SEM image of the s ame region in Fig. 2a. Th e agree- men t b et ween the tw o images is app aren t, ev en w ell outside the field-of-view of the original scanning area, thanks to the wide b eam jitter effectiv ely enlarging th e imaged area. Th e fidelit y of the reconstructed ob ject confirms the robustness of the algorithm and the v alid- it y of the retrieved prob es, whose comp onen ts are repr esented in Fig.s 3a-j. Though eac h comp onen t’s con tribution to eac h p u lse v aries, a qualitativ e indication of their relativ e w eight is giv en by the sin gu lar v alues obtained through tr uncated singu lar-v alue decom- p osition (SVD) and annotated on eac h comp onent (cf Fig.s 3a-j). When bac k-propagated to the virtual secondary source p lane lo cated at the mid -p oin t b et ween the pair of KB mirrors, 1 . 48 m up stream from th e inte r actio n plane (Fig.s 3k-t) the comp onents exhibit the exp ected in tensit y distribution, with the in tensit y dropp ing to negligible v alues outside of the main pup il. IUCr macros version 2.1.11: 2019/01/ 14 7 a 6 2 .0 % 2 0  m b 9 . 7 % c 7 . 0 % d 5 . 2 % e 4 . 0 % f 3 . 4 % g 2 . 9 % h 2 . 5 % i 2 . 0 % j 1 . 4 % k 4 m m l m n o p q r s t 0 Fig. 3. (a-j) 10 comp onents of reconstructed pr ob es obtained via OPRP r econstruction. Normalised singular v alues are annotated on eac h comp onent. (k-t) Bac k-p ropagatio n of same comp onen ts to the pupil plane, neglecting sp h erical w a ve term. Amplitude is mapp ed to bright n ess and phase to hue according to the color wheel in (a); in tensity scale is relativ e. Fig. 4. (a-d) Normalized amplitud e of 4 of the N = 937 retriev ed prob es obtained via OPRP reconstruction. (e) Amplitude of prob e retriev ed f r om Hartmann s ensor data. (f- j) Bac k-propagation of (a-e) to th e virtual p upil plane, n eglecting sph erical wa v e term. IUCr macros version 2.1.11: 2019/01/ 14 8 The normalized amplitude of four selected prob es is shown in Fig.s 4a-d as a repre- sen tativ e sample of the full stac k of N = 937 retriev ed prob es which is av ailable as a video in th e S upplement ary Material. F or comparison p urp oses, the amplitude of a prob e retriev ed s tarting from Hartmann sensor d ata is sh o wn in Fig. 4e. T he Hartmann sensor routinely a v ailable at the b eamline for w a vefron t sensing (Raimondi et al. , 2013) was op er- ated within the same exp erimental conditions as those of the ptyc hograph y exp eriment, although it did not sample th e same pulses. Figures 4f-j sho w the amplitude of the same w av efron ts (Fig.s 4a-e) after they ha ve b een n u merically bac k-prop agated to the virtual secondary sou r ce plane. Pulse-to-pulse v ariations can b e obser ved b oth at the samp le plane and at the virtual secondary sour ce plane. Using the full stac k of retriev ed prob es, it is p ossib le to gather v aluable statistica l infor- mation, such as v ariations in in tensit y and b eam p oin ting. Figure 5a illustrates the flu c- tuations of the total in tensity found for eac h prob e relativ e to the median total intensit y , rev ealing significant v ariations, w ith a relativ e stand ard deviation of 0 . 4. Figure 5b shows the radial displacemen t of the cen ter of mass of eac h prob e relativ e to the cen ter of the detector, rev ealing a median relativ e displacemen t of 8 µ m, confirming the p resence of b eam p oin ting v ariations. IUCr macros version 2.1.11: 2019/01/ 14 9 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 I/ I ( r e l a t i v e i n t e n si t y ) 0 2 0 1 0 2 0 3 0 4 0 5 0 r a d i a l d i sp l a c e m e n t ( μ m ) 0 2 5 5 0 a b 2 0 3 0 4 0 5 0 0 2 4 Fig. 5. Histograms of (a) in tensity of eac h prob e relativ e to the median intensit y) and (b ) radial displacemen t of the cen ter of mass of eac h prob e relativ e to cen ter of the detector. Inset in (b) is a rescaled version of a p ortion of th e same histogram. 3 mm b 0 a 30 µm z x z y Fig. 6. Horizon tal (a) and vertic al (b) sections of pt ychog r aphic reconstruction of main comp onen t propagated around the fo cal p osition ( ± 20 mm). Image scaling is different in its t w o d im en sions according to th e scale bars in (a). Amp litude is mapp ed to b r igh tness and phase to hue acco r ding to the color wheel in (a). IUCr macros version 2.1.11: 2019/01/ 14 10 W e n u merically propagated the main comp onent – as it is the most represent ative part of all retriev ed prob es – downstream and upstream fr om ob ject plane to inv estiga te th e region around the fo cal p osition. Sections of the p ropagated main comp onen t are sho wn in Fig. 6 and can b e compared with the unpropagated reconstru cted main comp onen t from Fig. 3a. The dotted line in Fig. 6 indicates the sample plane; the fo cal p lane has b een estimated to b e some 2 mm further ups tr eam, as the p lane at w h ic h the b eam size is minimized. 4. Conclusions W e h a v e demonstrated the first fully p ulse-to-pulse pt ychog r aphic w av efron t reconstruction for a FEL instrument in its nativ e imaging state. Ou r approac h pr o vides the complex- v alued wa v efront of eac h p ulse directly at the sample plane, in its near-focus p osition. As a lensless microscop y tec hnique, its ac hiev able resolution is limited only b y fl ux and detector n u m erical ap erture. The full set of 937 retriev ed prob es rev eals pulse-to-pulse v ariations, providing statistical information on FEL b eam fluctuations and d irect ins ight in to FERMI’s p erformance. The retriev ed prob es also confirm the elongated shap e of the b eam at and around the fo cal p osi- tion, as exp ected from p revious exp erimen ts carried out at the same b eamline (Raimondi et al. , 2013 ). Pt yc h ographic w av efron t c h aracterisati on brings some b enefits compared to grating- based metho ds (S c hneider et al. , 2018; Liu et al. , 2018), which reco ve r information far from the fo cal p osition, tend to underestimate the intensit y in th e tails of the p ow er distri- bution and are limited by grating fabrication. T he metho d has also prov en effectiv e even in the presence of signifi cant setup vibrations – another adv an tage o ver Hartmann and IUCr macros version 2.1.11: 2019/01/ 14 11 grating-based w av efron t sensing, whic h are mostly insens itive to b eam p osition v ariations. Pulse-to-pulse pt ychograph y is primarily exp ected to b enefit w av efron t c h aracterization exp erimen ts at FELs, contributing to the dev elopmen t of FEL and FEL-based science. Its application can b e extended to other imaging exp erimen ts, whic h w ould b enefit f r om the relaxation of the single-illumination prob e constraint. App endix A Exp erimen tal setup The pt yc hography exp erimen t w as carried out at the Diffraction and Pro jection Imaging (DiProI) b eamline (Cap otondi et al. , 2013; Cap otondi et al. , 2015) at FERMI, EUV and soft X-ra y seeded FEL, using the FEL-2 line (Allaria et al. , 201 2; Allaria et al. , 2013; Allaria et al. , 2015). T he v ertical and horizon tal b endable KB mirrors h ad a fo cal length of 1 . 75 m and 1 . 2 m, resp ectiv ely (Raimond i et al. , 2013). Th e sample (Xradia X30-30-2 ) w as a 110 nm Si 3 N 4 mem br ane w ith a 200 nm thic k Au test pattern dep osited on top. A 3-axis tran s latio n stage w as used to translate and scan the samp le within a v acuum c hamb er. The in -v acuum CCD camera was a P I -MTE:204 8B with 2048 × 2048 pixels, 13 . 5 µ m eac h. T o decrease readout time to 2 s, only the in tensities collected by the cen tral 1000 × 100 0 pixels were recorded. As readout frequ en cy was low er than the FEL’s 10 Hz rep etition rate, a f ast sh u tter wa s used to prev ent more than one pulse from con tributing to eac h detector reading. Due to furth er o v erhead, the acquisition r ate w as effectiv ely reduced to 0 . 2 Hz, i.e. only one eve r y 50 FEL pulses w as recorded. IUCr macros version 2.1.11: 2019/01/ 14 12 The b eam was attenuat ed with a combination of atten uators. A 6 m long gas cham b er w as filled with 2 . 8 × 10 − 2 m bar of N 2 and complemen ted with solid Zr (600 nm) and Al (200 n m) atten u ators. Thus atten u ated, pulses remained well b elo w the sample’s damage threshold, give n by Au m elting dose of 0 . 4 eV p er atom (Da vid et al. , 2011). App endix B Data acquisition and pro cessing Pt yc h ographic data were collected using three 25 × 25 µ m 2 spiral scans in the x - y plane, with a step size of 2 . 5 µ m for a total of 101 p ositions eac h. Five single-pulse diffraction patterns w ere collected at eac h scanning p osition, for a total of 1515 frames o ve r the 3 spiral scans. The cen ter of the last sp iral scan wa s translated in y to extend the ov erall scanned area whic h is highlight ed by th e rectangles in Fig.s 7a,b. The collecte d d iffr actio n patterns were binn ed b y a factor of 2 and then padded to a size of 512 × 512 pixels in order to decrease compu tational cost. Each fr ame wa s th en dark-su btracted. Detector coun ts w ere thresholded to a v alue of 0 in order to remov e un ph ysical negativ e coun ts and conv erted in to units of photon counts. IUCr macros version 2.1.11: 2019/01/ 14 13 a b c 10 µm y x 10 µm Fig. 7. (a) S EM image of the Siemens star test p attern; scalebar in (b). (b) Prob e p ositions reco v ered via our p osition correction algo r ithm. The area co v ered by the motor p ositions used for the p t yc hographic scan is annotated w ith rectangles in (a) and (b). (c) Positio n correction for eac h scanning p oint; null correction (0,0) at cen ter. The main axis of suc h t w o-dimens ional correction distribu tion is annotated rev ealing a dominan t vibration comp onen t along the x axis. A wid e jitter of the p hoton b eam p osition was found to ha ve o ccurred throughout d ata acquisition. Ascrib ed to vibrations in the optics up s tream from the exp erimental setup, the jitter w as of such a large amplitude – tens of microns – th at the scanning p ositions recorded b y the sample motors w ere essen tially u n usable. Most k n o wn p osition refine- men t approac hes within ptyc hographic algorithms (Guizar-Sicai r os & Fien u p, 2008; Maiden et al. , 2012 ; Bec kers et al. , 2013; Zhang et al. , 2013 ; T ripathi et al. , 2014 ) ha v e b een designed to accoun t for minor deviations from the exp ected p ositions, t ypically smaller than the scan- ning step size. Here, a customized approac h was imp lemen ted to accommodate f or such a wide jitter. Coarse p ositions w ere fir st obtained by cross-correlating the recorded diffrac- tion patterns with s im ulations based on the in teraction of a sim u lated p rob e and an ob ject mo delled fr om the av ailable high-resolution im age of the test pattern (cf Fig. 7a). T h e coarse p ositions were th en further refined with a pu rp osely-designed p osition refin emen t algorithm (Ods tr ˇ cil e t al. , 2018). Diffraction patterns corresp onding to p ositions far fr om IUCr macros version 2.1.11: 2019/01/ 14 14 the int en d ed field-of-view w ere discarded, as we ll as others for whic h p osition refinement failed, effectiv ely red u cing the d ataset to 937 f rames. Figure 7b s h o ws the corr ected and refin ed p ositions, along w ith the int end ed rectangular scanning area. T he deviation of th e retrieve d scanning p ositions to the nominal ones is represent ed in Fig. 7c where the main axis of the vibr ation d istribution is annotated, rev ealing a d omin an t horizont al comp onent asso ciated to a standard deviation of 9 . 2 µ m. Although the coarse p osition correction was only p ossible than k s to the in-dep th c har- acterizat ion of the test pattern prior the exp eriment, the deve lop ed p osition refin emen t algorithm is exp ected to b enefit sev eral other exp eriments p erformed at FELs which are affected b y int r insic p oin ting instabilit y , b eside minor vibrations of the optics and sample stages. Giv en the exp eriment geomet r y and d etecto r sp ecifications, the ac hieved p ixel size in the pt yc h ographic reconstruction was 162 nm. The initial illumination fun ction was p ro- duced by n umerical bac k-propagation of the m ean diffraction pattern. The initial ob ject w as assigned a uniform unit trans mission fun ction. The pt ychographic algorithm ran 200 iterations of DM, follo wed by 800 of ML refin emen t. App endix C Reconstruction algorithm The pt yc hographic wa v efront characte r ization approac h used within this work for single- pulse in vestig ation is deriv ed from OPRP (Od strcil et al. , 2016). It reco vers a different prob e P j for eac h of the N diffractio n patterns I j recorded d u ring a ptyc h ographic scan, IUCr macros version 2.1.11: 2019/01/ 14 15 with the frame index j v arying b et we en 1 and N . The SVD step is added at the end of ev ery iteration of the ptyc hographic reconstruction algorithm and generates the p rob es’ main prin cipal comp onen ts or “mo des”. Giv en the complex matrix P whose columns con tain estimates of the ind ivid ual pr ob es P j , applying SVD to P leads to P = U Σ V ∗ where V ∗ denotes the Hermitian transp osition of V and b oth U and V are unitary matrices suc h that U U ∗ = U ∗ U = I and V ∗ V = V V ∗ = I with I as the identi ty matrix. By m u ltiplying P with its Herm itian transp ose P ∗ , one gets P ∗ P = V Σ ∗ U ∗ U Σ V ∗ = V (Σ ∗ Σ) V ∗ (1) with P ∗ P as a Hermitian matrix and (Σ ∗ Σ) as a d iagonal matrix. This is equiv alen t to th e eigen v alue p roblem ( P ∗ P ) V = V (Σ ∗ Σ) (2) so that the non-zero elements on th e d iagonal of Σ corresp ond to the square ro ots of th e eigen v alues of P ∗ P . The solution to this problem w ithin th e p tyc hographic algorithm is implement ed as a truncated diagonalisat ion: the N eigen v alues and main k eigen v ectors ˆ V can b e retrieved, with k < N and ˆ V denoting truncation of V . Ap plying this in the S VD step, a set of k orthogonal comp onents M is generated via M = P n ˆ V = U ˆ Σ ˆ V ∗ ˆ V = U ˆ Σ wher e P n denotes the prob e matrix P at the n -th iteration. The obtained comp onen t matrix M is then u sed to generate the u p dated prob es P n +1 = M ˆ V ∗ = U ˆ Σ ˆ V ∗ . W e are thankful to N. Mahne, M. Zangrand o and L. Raimondi from the P ADReS group at FERMI for their con tribu tion to the exp erimen t leading to these results. W e ac kno wl- edge the u se of the IRIDIS High P erformance C omputing F ac ility , and asso ciated sup p ort IUCr macros version 2.1.11: 2019/01/ 14 16 services at the Univ ersity of Southampton, in the completion of this wo r k. The researc h leading to these resu lts h as receiv ed fu nding from the Europ ean Communit y’s Sev enth F ramework Programme (FP7/2007-20 13) und er gran t agreement s n. 279753 and n. 312284 and from Diamond Light Source Limited, the Swedish R esearch Council, the Knut and Alice W allen b erg F ound ation, the S wedish F oundation for Strategic Research and the Sw edish F oun dation for I nternational Co op eration in Researc h and Higher Education (STINT). References Allaria, E., Appio, R., Bada no , L., Barletta, W. A., Bassa nese, S., Biedron, S. 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