Fourier-transform Ghost Imaging with Hard X-rays

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1 Fouie-ansfom Ghos Imaging wih Had X-ays Hong Yu 1, Ronghua Lu 1, Shensheng Han 1*, Honglan Xie, Guohao Du, Tiqiao Xiao, Daming Zhu 3,4 1 Shanghai Insiue of Opics and Fine Mechanics, Chinese Academy of Science, Shanghai, 01800, China Shanghai Insiue of Applied Physics, Chinese Academy of Science, Shanghai, 01800, China 3 Univesiy of Science and Technology of China, Hefei, 3007, China 4 Univesiy of Missoui-Kansas Ciy, Kansas Ciy, Missoui 64110, USA Absac: Knowledge gained hough X-ay cysallogaphy foseed sucual deeminaion of maeials and gealy faciliaed he developmen of moden science and echnology in he pas cenuy. Aomic deails of sample sucues is achievable by X-ay cysallogaphy, howeve, i is only applied o cysalline sucues. Imaging echniques based on X-ay coheen diffacion o zone plaes ae capable of esolving he inenal sucue of non-cysalline maeials a nanoscales, bu i is sill a challenge o achieve aomic esoluion. Hee we demonsae a novel lensless Fouie-ansfom ghos imaging mehod wih pseudo-hemal had X-ays by measuing he second-ode inensiy coelaion funcion of he ligh. We show ha high esoluion Fouie-ansfom diffacion paen of a complex ampliude sample can be achieved a Fesnel egion and he ampliude and phase disibuions of a sample in spaial domain can be eieved successfully. The mehod of lensless X-ay Fouie-ansfom ghos imaging exends X-ay cysallogaphy o non-cysalline samples, and is spaial esoluion is limied only by he wavelengh of he X-ay, hus aomic esoluion should be ouinely obainable. Since highly coheen X-ay souce is no equied, compaing o convenional X-ay coheen diffacion imaging, he mehod can be implemened wih laboaoy X-ay souces, and i also povides a poenial soluion fo lensless diffacion imaging wih femions, such as neuon and elecon whee he inensive coheen souce usually is no available. Since Max Laue discoveed X-ay diffacion in cysals in 1914, X-ay cysallogaphy has become a poweful ool in exploing and analyzing he inenal sucues of complex maeials, such as biomolecula sucues and nanomaeials 1-3. The esoluion of X-ay cysallogaphy is only limied by he wavelengh of he X-ay, which povides he oppouniy o visualize he aomic deails of cysalline sucues. Howeve, sucue infomaion of many impoan molecula maeials, such as membane poeins, is sill ou of each, because hese maeials ae difficul o gow ino macoscopic cysals. Fuhemoe, wih he apid developmen of nanoscience and biology, i has become an ugen need o obain aomic esoluion images of he inenal sucue of samples in hei naual saes insead of in cysals. In 1999, Coheen diffacion imaging (CDI) mehod 4-6 was poposed o exend X-ay cysallogaphy o allow imaging non-cysalline sucues in nanoscale by illuminaing he samples wih coheen X-ays and ecoding he diffacion paens in fa-field. Fesnel CDI 7,8 and pychogaphy echnique 9,10 wee also poposed o cicumven he ininsic esicion of sample size in classical CDI. Neveheless, due o he equiemens fo high coheence and bighness, synchoon adiaion o X-ay fee elecon lase souces is sill essenial o X-ay CDI applicaions, and high esoluion imaging using laboaoy X-ay souce wih CDI echniques emains o be achieved. Mos of convenional imaging mehods ae based on deecion of inensiy disibuion of ligh fields, i.e. he fis-ode coelaion of he ligh. In fac, imaging in boh eal and ecipocal space can * sshan@mail.shcnc.ac.cn

2 be ealized wih hemal ligh hough ghos imaging echnique, a phenomenon fis obseved in quanum egime wo decades ago 11,1, by measuing highe ode coelaion of ligh fields. Diffeen fom convenional mehods, in a ypical ghos imaging sysem, he ligh field passing hough o efleced by a sample is ecoded only wih a non-spaially esolving deeco(i.e. a bucke o poin deeco), and he sample s infomaion is acquied fom he second-ode inensiy flucuaion coelaion of he scaeed and unscaeed lighs. Ghos imaging has been poved and demonsaed wih classical visible hemal ligh and applied quickly in emoe sensing, phoolihogaphy, supe esoluion imaging, single-pixel hee dimensional camea, ec A lensless X-ay Fouie-ansfom ghos imaging scheme 15,3 wih spaially incoheen illuminaion, whee he Fouie-ansfom diffacion paen of he sample can be acquied a Fesnel egion, had also been poposed o achieve he same diffacion paen as in CDI. Theefoe Fouie-ansfom ghos imaging povides a possibiliy o achieve aomic esoluion images of non-cysalline samples wih widely accessible laboaoy incoheen X-ay souces. Howeve, a beam splie is needed in he scheme o geneae wo copies of he inciden ligh field, which is difficul fo X-ay. In his pape, we epo fo he fis ime he success of an expeimen which demonsaes he feasibiliy of pefoming had X-ay Fouie-ansfom ghos imaging (FGI) using a navel expeimenal appoach. The expeimen was based on he same pinciple of visible ligh FGI which is illusaed in Fig.1(a). A ligh beam fom a spaially incoheen souce is divided ino wo beams, a esing beam and a efeence beam, afe passing hough a beam splie. The ligh in he esing beam passes hough a sample a a disance d 1 and hen is ecoded by a poin deeco D posiioned a a disance d fom he sample. The efeence beam does no pass hough he sample a all, bu is inensiy disibuion afe passing hough a disance d is ecoded by a spaially esolved panel deeco D. Fig. 1. Illusaion of he expeimenal seup fo FGI wihou a beam splie. (a) is he pinciple of lensless FGI, and (b) is he seup of a lensless FGI sysem wihou a beam splie: Fo a conollable pseudo-hemal ligh souce, when he sample is inseed ino he ligh beam as in mode 1, each pixel of he panel deeco can be consideed as a poin deeco D in he esing beam of FGI, while when he sample is moved ou of he beam as in mode, he panel deeco ecods he inensiy disibuion wihou passing hough he sample as he efeence beam deeco D of FGI, if only he wo mode measuemens ae compleed in one pseudo-coheen duaion deemined by he oaing speed of he gound glass.

3 As he deivaions deailed in efeence [15] show ha, unde he condiion whee d1 + d = d, he coelaion funcion beween he inensiy flucuaion of efeence beam Δ I( x) and ha of he esing beam Δ I ( x ) measued by he wo deecos is diecly elaed o he modulus of he Fouie ansfomaion of he sample s ansmiance, and such a elaion can be expessed as d I π ( x x) ΔI( x) Δ I( x) = T( ), (1) λ λd whee Δ Ik( xk) = Ik( xk) < Ik( xk) >, k = o, x and x ae he coodinaes a he deeco planes in he efeence and esing pahs especively, λ is he wavelengh, I 0 is he inensiy of he π ( x x) inciden ligh, T( ) is he Fouie ansfomaion of he sample s ansmiance. Real space λd image of he sample can be eieved fom π ( x x) T( ) λd as in convenional X-ay CDI. Thus, he sample can be imaged by ecoding he incoheen spaial inensiy disibuion of he beam wihou passing hough he sample and hen coelaed wih he signal deeced by a single pixel deeco placed behind he sample. Such an imaging mehod emoves he need fo a beam sop and avoids missing low-fequency daa in he diffacion paens as in adiional X-ay CDI, which has been inensively sudied using he visible pseudo-hemal ligh souce in he pas decade 17,33. Howeve, o es he FGI in he had x-ay egime is challenging. Fo had X-ays, hee is no pefec beam splie which can poduce win beams as he case of visible ligh 3. To cicumven such a difficuly, we noice ha he condiion d 1 + d = d is saisfied in he scheme. So, fo conollable pseudo-hemal ligh souce, ahe han spliing he hemal ligh ino wo beams as in Fig.1(a), we can use an equivalen scheme as shown in Fig.1(b), which uses only one spaially incoheen pseudo-hemal ligh beam and a fixed panel deeco by shuling he sample in and ou of he beam in one pseudo-coheen duaion (duing which he ligh souce is sable). When he sample is inseed ino he beam, he signals deeced by a single pixel of he panel deeco seve as ha deeced in he poin deeco in he esing beam, when he sample is moved ou of he beam, he signals deeced seve as ha deeced in he panel deeco in he efeence beam, so boh he inensiy flucuaion Δ I ( x ) and Δ I( x) can be acquied by he same panel deeco if only he wo measuemens ae compleed in one pseudo-coheen duaion. The expeimen was pefomed on he 13W beamline a Shanghai Synchoon Radiaion Faciliy (SSRF) which is dedicaed o X-ay imaging and biomedical applicaions. Fig.(a) shows he expeimenal seup. A pseudo-hemal X-ay souce 34-36, which can geneae a conollable chaoic X-ay speckle paen flucuaion o emulae he behavio of a spaially incoheen X-ay souce, was used o illuminae he sample. The pseudo-hemal X-ay souce is poduced by a monochomaic X-ay beam passing hough a sli aay and a moveable gold film deposied on a Si3N4 subsae. The monochomaic X-ay beam was poduced by passing he X-ays emied fom he synchoon souce hough a double cysal monochomao wih an enegy esoluion ΔE/E The flux of he x-ay was phoons/mm /sec and he enegy was ceneed a 1.1 kev(0.1 nm wavelengh). The sli aay is posiioned in he opical pah of he X-ay beam, and each sli of he aay has a dimension less han o equal o he X-ay s coheen aea a he sli aay plane (50um 10um), so ha each of he ansmied X-ay sub-beam passing hough a sli is spaially coheen. The gold film wih andomly disibued holes of diamee<1um was mouned on a compue-conollable anslaional device and

placed closely behind he sli aay, he deph of he holes ae designed o be λ /( n 1) =.7um o fom a phase diffeence of π beween he aea wih and wihou holes fo 1.1 kev had X-ay.

4 placed closely behind he sli aay, he deph of he holes ae designed o be λ /( n 1) =.7um o fom a phase diffeence of π beween he aea wih and wihou holes fo 1.1 kev had X-ay. Afe he bundle of spaially coheen X-ay sub-beams fom he sli aay passing hough he gold film, chaoic disibued X-ay speckle paens ae poduced because of he spaially sochasic inefeence of he andomly modulaed spaial coheen X-ay sub-beams fom he gold film. The size of he gold film is much lage han he whole beam coss-secion of he monochomaic X-ays, hus when he gold film is moved ansvesely by he anslaional device o make he diffeen pa of he film be illuminaed, a pseudo-hemal X-ay beam wih chaoic flucuaing inensiy disibuions was poduced and seved as he incoheen X-ay souce in ou expeimen. The CCD camea wih effecive pixel size of 0.37um*0.37um was placed 43cm downseam fom he gold film. The expeimenal sample was placed on a conollable anslaional sage, which can move he sample in and ou of he pseudo-hemal X-ay beam as shown by he ed aow in Fig.(a), fo pefoming Δ I ( x ) and Δ I ( x ) measuemens. In one measuemen, a pai of X-ay signals, i.e. he inensiy flucuaion of he esing beam Δ I ( x ) and ha of he efeence beam Δ I ( x ), was acquied as shown in Fig.(b), and he gold film was igged o move ansvesely again afe each of he measuemens was compleed. We should menion ha he dif of he sample posiion elaive o he illuminaing pseudo-hemal X-ay beam inoduced by he shuling was almos ineviable, bu i makes lile influence o FGI sysem as demonsaed heoeically and expeimenally in efeence [37]. Fig.. Expeimenal seup fo X-ay FGI using a pseudo-hemal X-ay souce is shown in (a). A monochomaic X-ay beam passes hough a sli aay and a moving film o geneae a conollable pseudo-hemal X-ay speckle paen flucuaion which is used in he expeimen as he pseudo-hemal X-ay souce. The sample is moved in and ou of he beam o ge he inensiy flucuaion Δ I ( x ) and Δ I ( x ) especively as equied by he scheme of FGI wihou a beam splie. Fig. (b) shows he opical micoscope image of he sample and an example of inensiy disibuion paen pais Δ I ( x ) and Δ I ( x ) acquied in X-ay FGI expeimen. The sample in ou expeimen was a.um hick gold film wih five slis on a Si3N4 subsae. The slis wee sepaaed by d = 3um and he widh of each sli was 1um. Fig.(b) shows he opical sli micoscope image of he sample. Since he wavelengh of he pseudo-hemal X-ay souce was 0.1nm, he inensiy aenuaion and phase diffeence beween he slis and he suounding gold aea was 53% and 0.8π, especively. The disance fom sample o CCD camea is 33cm and he widh of he sample illuminaed by he 0.1nm pseudo-hemal X-ays is 13um, so he fa-field diffacion condiion ( D / λ = 13 um /0.1nm= 1.69m, D is he oal widh of he five slis) is no saisfied. Fo FGI pupose, signals fom only one fixed single pixel in he inensiy paen, ecoded while he imaging example was placed in he beam, wee needed o consuc he coelaion funcion. Fo convenience,

5 we ecoded he enie inensiy paen of he beam passing hough he sample. An example of he inensiy disibuion paen pais is shown in Fig. (b). Obviously, he X-ay inensiy paens display feaueless andom disibuions, and no diffacion paens of he sample can be diecly obseved. Fig. 3. Diffacion paens of he sample obained wih X-ays of 0.1nm wavelengh. (a) is he Fouie-ansfom diffacion paen of he sample s ansmiance obained by X-ay FGI, (c) is he Fouie-ansfom diffacion paen of he squaed modulus of he sample s ansmiance obained in X-ay FGI, (e) is he inensiy disibuion obained by illuminaing he sample diecly wih synchoon X-ays. The ed lines in (b), (d) and (f) ae he coss-secion cuves of (a), (c) and (e), especively. The blue lines in (b) and (d) ae he coesponding numeical esuls obained by Faunhofe diffacion inegal. The Fouie-ansfom diffacion paen of he complex ampliude sample was obained by calculaing he coelaion beween he inensiy flucuaions of he esing beam Δ I ( x ) and he efeence beam Δ I ( x ) following Eq.(1). Howeve, o impove he sampling efficiency in he calculaion, we applied a econsucion algoihm making use of spasiy consains of image 38 in econsucing he Fouie-ansfom diffacion paens. Fig.3(a) is he sample s diffacion paen obained by X-ay FGI wih 84 pais of measuemens daa used in econsucion calculaion. The coss-secion cuve of Fig.3(a) is shown by he ed line in Fig.3(b), and he peak spacing of he ed line in Fig.3(b) is abou 11.1um(0.37um/pixel*30 pixels=11.1um), which is in ageemen wih he heoeical value of he peak spacing( λ d / d = 0.1nm 33 cm /3um = 11um ) pediced by Eq.(1). The sli blue line in Fig.3(b) shows he numeical esul of he sample s Fouie ansfomaion, and i agees well wih he expeimen esul. Theefoe, he sample s Fouie-ansfom diffacion paen was

6 obained a Fesnel egion by X-ay FGI, which is diffeen fom he case in X-ay CDI, whee he sample s Fouie-ansfom diffacion paen should be obained a fa-field. By calculaing he auo-coelaion of he inensiy flucuaion of he esing beam Δ I ( x ), he Fouie-ansfom diffacion paen of he squaed modulus of he sample s ansmiance can also be obained 39. The Fouie-ansfom diffacion paen of he squaed modulus of he sample s ansmiance obained in X-ay FGI and he coesponding coss-secion cuves ae shown in Fig.3(c) and Fig.3(d), especively. As Fig.3(d) shows, he expeimen esul agees well wih he numeical Fouie ansfomaion esul. Thus, he ampliude and phase infomaion of he complex ampliude sample wee obained sepaaely in ou X-ay FGI expeimen. Fo efeence, he X-ay inensiy disibuion ecoded by he CCD camea when he sample was diecly illuminaed by he monochomaic X-ay beam emied fom he synchoon souce hough he double cysal monochomao is shown in false-colo epesenaion as Fig.3(e). Fig.3(f) shows he coss-secion cuve of Fig.3(e). By compaing Fig.3(e) wih Fig.3(a) and Fig.3(c), i can be found ha he paen obained when illuminaing he sample wih monochomaic X-ays is appaenly diffeen fom he Fouie-ansfom paens obained in X-ay FGI. Using a wo-sep phase-eieval image econsucion pocess based on FGI 4, in which fisly he ampliude pa of he sample s ansmiance is eieved fom he Fouie-ansfom paens in Fig.3(c), hen combining he eieved ampliude pa of he sample s ansmiance wih he Fouie-ansfom paen in Fig.3(a), he phase pa of he sample s ansmiance was eieved. The eieved ampliude and phase disibuions of he sample s ansmiance ae shown in Fig.4(a) and Fig.4(b), especively. The maximum spaial fequency used in he econsucion can be calculaed as q max 0.37 um pixel *300pixels =, so he pixel size in he eieved image is λd 1 q max = 0.97um, and he sepaae disance beween he slis in he eieved image is.97um(0.97um/pixel*10 pixels), which is in compleely ageemen wih he spaial feaue of he sample. Fig. 4. The sample s ampliude disibuion (a) and phase disibuion (b) in spaial domain eieved fom he Fouie-ansfom diffacion paens obained in X-ay FGI. Ou expeimenal esuls demonsaed fo he fis ime ha Fouie-ansfom ghos imaging can be achieved using pseudo-hemal had X-ays. The diffacion paens ae qualified enough o eieve he ampliude and phase disibuions of he sample in spaial domain. The spaial esoluion of FGI wih incoheen X-ays is deemined by he maximum spaial fequency of he Fouie-ansfom diffacion paen, which means he spaial esoluion of lensless X-ay FGI is only limied by he

7 wavelengh, and povides he poenial o achieve aomic esoluion images of non-cysalline samples wih laboaoy X-ay souces. In summay, we have expeimenally demonsaed Fouie-ansfom ghos imaging wih pseudo-hemal had X-ays and high esoluion Fouie-ansfom diffacion paen of he sample has been achieved a Fesnel egion by measuing he second-ode inensiy coelaion of he lighs, he ampliude and phase disibuions of he sample in spaial domain have been eieved successfully. This mehod exends X-ay cysallogaphy o non-cysalline samples and, as been a lensless imaging scheme, he spaial esoluion of X-ay FGI is only limied by he wavelengh. An impoan feaue of X-ay FGI mehod is ha i does no ely on highly coheen X-ay souce o ealize X-ay diffacion imaging, heefoe i povides a feasible way o achieve aomic esoluion images of non-cysalline samples wih widely accessible laboaoy X-ay souces. Eliminaing he need fo inensive coheen souce, FGI of no only bosons bu also femions such as neuon and elecon, can also be expeced, so i povides a glimpse of possibiliy of evoluionizing he cuen neuon and elecon scaeing mehods widely used in eseaches in maeials sciences as well as biomedicine. Fuhemoe, he Fouie-ansfom diffacion paen is acquied fom coelaed calculaions, which emoves he need fo a beam sop and avoids missing low-fequency daa in he diffacion paens as in adiional X-ay CDI. Finally, he high fequency poion of he diffacion paen ha is ou of he CCD deecion aea can also be capued by using he inensiy disibuion paens deviae fom he cene poin in pais 39, which means he spaial esoluion of X-ay FGI sysems can be doubly enhanced. The wok is suppoed by he Hi-Tech Reseach and Developmen Pogam of China unde Gan Pojecs No.013AA1901 and No.013AA190, he Naional Naual Science Foundaion of China unde Gan Pojecs No , and he Shanghai Fundamenal Reseach Pojec No.09JC Shi, Y. A glimpse of sucual biology hough X-ay cysallogaphy. Cell 159, (014).. Fleuy, B. e al. Gold nanopaicle inenal sucue and symmey pobed by unified small-angle X-ay scaeing and X-ay diffacion coupled wih molecula dynamics analysis. Nano Le. 15, (015). 3. Kosynkin, D.V. e al. Longiudinal unzipping of cabon nanoubes o fom gaphene nanoibbons. Naue 458, (009). 4. Miao, J., Chaalambous, P., Kiz, J., Saye, D. Exending hemehodology of X-ay cysallogaphy o allow imaging of micomee-sized non-cysalline specimens. Naue 400, (1999). 5. Chapman, H.N. e al. Femosecond diffacive imaging wih a sof-x-ay fee-elecon lase. Na. Phys., (006). 6. Pfeife, M.A., Williams, G.J., Vaanyans, I.A., Hade, R., Robinson, I.K. Thee-dimensional mapping of a defomaion field inside a nanocysal. Naue 44, 63-66(006). 7. Williams, G.J. e al. Fesnel coheen diffacive imaging. Phys. Rev. Le. 97, 05506(006). 8. Abbey, B. e al. Keyhole coheen diffacive imaging. Na. Phys. 4, (008). 9. Rodenbug, J.M. e al. Had X-ay lensless imaging of exended objecs. Phys. Rev. Le. 98, (007). 10. Thibaul, P. e al. High-esoluion scanning X-ay diffacion micoscopy. Science 31, 379(008). 11. Belinskii, A.V., Klyshko, D.N. Two-phoon opics: diffacion, hologaphy, and ansfomaion of wo-dimensional signals. J. Exp. Theo. Phys. 78, 59-6(1994). 1. Piman, T.B., Shih, Y.H., Sekalov, D.V., Segienko, A.V. Opical imaging by means of wo-phoon quanum enanglemen. Phys. Rev. A 5, R349-R343(1995).

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Chapter 7. Interference

Chapter 7. Interference Chape 7 Inefeence Pa I Geneal Consideaions Pinciple of Supeposiion Pinciple of Supeposiion When wo o moe opical waves mee in he same locaion, hey follow supeposiion pinciple Mos opical sensos deec opical