Coherent Diffraction Imaging

Ross Harder aka “The Imposter” Acknowledgments: Prof. Ian Robinson (BNL) 34-ID-C Dr. Xiaojing Huang (BNL) Advanced Photon Source Dr. Jesse Clark (PULSE institute, SLAC Amazon) Prof. Oleg Shpyrko (UCSD) Dr. Andrew Ulvestad (ANL – MSDTesla) https://tinyurl.com/y2qtrz7c Dr. Ian McNulty (ANL – CNM MaxIV) Dr. Junjing Deng (ANL – APS)

Non-compact samples (Ptychography) Compact Objects (CDI) research papers implies an effective degree of transverse coherence at the oftenCOHERENCE associated with interference phenomena, where the source, as totally incoherent sources radiate into all directions mutual coherence function (MCF)1 (Goodman, 1985). À r ; r ; E Ã r ; t E r ; t 2 The transverse coherence area ÁxÁy of a synchrotron ð 1 2 Þ¼ ð 1 Þ ð 2 þ Þ ð Þ source can be estimated from Heisenberg’s uncertainty prin- plays the main role. It describes the correlations between two ciple (Mandel & Wolf, 1995), ÁxÁy h- 2=4Áp Áp , where x y complex values of the electric field E Ã r1; t and E r2; t at Áx; Áy and Áp ; Áp are the uncertainties in the position and ð Þ ð þ Þ x y different points r1 and r2 and different times t and t .The momentum in the horizontal and vertical direction, respec- brackets denote the time average. þ tively. Due to the de Broglie relation p = h- k,wherek =2=, Whenh weÁÁÁ consideri propagation of the correlation function the uncertainty in the momentum Áp can be associated with of the field in free space, it is convenient to introduce the the source divergence Á, Áp = h- kÁ,andthecoherencearea cross-spectral density function (CSD), W r1; r2; ! , which is in the source plane is given by defined as the Fourier transform of the MCFð (MandelÞ & Wolf, 1995), 2 1 ÁxÁy : 1 4 Á Á ð Þ W r1; r2; ! À r1; r2; exp i! d; 3 x y ð Þ¼ ð Þ ðÀ Þ ð Þ where ! Visibilityis the frequency of fringes is of a the direct radiation. measure By of the definition, coherence when of a beam. Substituting typical values of the source divergence at a third- R the twoIf points beam isr coherentand r acrosscoincide, the spacing the CSD of the represents slits a Fourier the Transform generation synchrotron source (Balewski et al.,2004)into 1 2 spectral densityof the slit of structure the radiation is observed field, downstream. equation (1), we find the minimum transverse coherence 2 length at the source to be about a few micrometers. With the S r; ! W r; r; ! : 4 source sizes of tens to hundreds of micrometers (Balewski et ð Þ¼ ð Þ ð Þ The normalized CSD is known as the spectral degree of al.,2004),itisclearthatpresentthird-generationX-ray coherence (SDC), sources have to be described as partially coherent sources.

Ausefulmodeltodescribetheradiationpropertiesof W r1; r2; ! r1; r2; ! ð Þ 1=2 : 5 partially coherent sources is the Gaussian Schell-model ð Þ¼ S r ; ! S r ; ! ð Þ (GSM) (Mandel & Wolf, 1995). This model has been applied ð 1 Þ ð 2 Þ for the analysis of the radiation field generated by optical For all values of r ; r and ! ÂÃthe SDC satisfies r ; r ; ! 1. 1 2 j ð 1 2 Þj lasers (Gori, 1980), third-generation synchrotron sources (see, The modulus of the SDC can be measured in interference for example, Howels & Kincaid, 1994; Vartanyants & Singer, experiments as the contrast of the interference fringes (Singer 2010, and references therein) and X-ray free-electron lasers et al.,2008,2012;Vartanyantset al.,2011). (Singer et al.,2008,2012;Rolinget al.,2011;Vartanyantset al., To characterize the transverse coherence properties of a 2011). The problem of propagation of partially coherent wavefield by a single number, the global degree of transverse radiation through thin optical elements (OEs) in the frame of coherence can be introduced as (Saldin et al., 2008; Vartany- GSM has been widely discussed in optics (Turunen & Friberg, ants & Singer, 2010) 1986; Yura & Hanson, 1987). However, the propagation of W r ; r ; ! 2 dr dr partially coherent radiation through the focusing elements 1 2 1 2 ! j ð Þj 2 : 6 with finite apertures has not been considered before. For ð Þ¼ S r; ! dr ð Þ R ð Þ X-ray beamlines at third-generation synchrotron sources such According to its definition theÂÃR values of the parameter ! lie focusing elements are especially important. In this work we in the range 0 ! 1, where ! =1and!ð Þ=0 propose a general approach to describe propagation of characterize fully coherentð Þ and incoherentð Þ radiation,ð respec-Þ partially coherent radiation through these beamlines. tively. The paper is organized as follows. We start with a short In the following we will apply the concept of correlation introduction to the optical coherence theory with special focus functions to planar secondary sources (Mandel & Wolf, 1995), on third-generation synchrotron radiation sources in 2. where the CSD of the radiation field is given in the source 3describesthepropagationofpartiallycoherentX-rayx plane at z =0withthetransversecoordinatess, radiationx through thin focusing elements. Diffraction-limited 0 W s ; s ; z ; ! .ThepropagationoftheCSDfromthesource focus and infinite apertures are described in 4and 5. In 6 1 2 0 planeð at z toÞ the plane at a distance z from the source is coherence properties of the focused X-ray beamsx atx PETRAx 0 governed by the following expression (Mandel & Wolf, 1995), III are analyzed. The paper is concluded with a summary and outlook. W u ; u ; z; ! ð 1 2 Þ¼ W s ; s ; z ; ! PÃ u ; s ; ! P u ; s ; ! ds ds ; 7 ð 1 2 0 Þ zð 1 1 Þ zð 2 2 Þ 1 2 ð Þ 2. Coherence: basic equations where WR u ; u ; z; ! is the propagated CSD in the plane z, ð 1 2 Þ 2.1. Correlation functions and propagation in free-space and Pz u; s; ! is the propagator. The integration is performed in the sourceð Þ plane. For partially coherent X-ray radiation at The theory of partially coherent fields is based on the treatment of correlation functions of the complex wavefield 1 Here we restrict ourselves to stationary radiation fields, which are generated (Mandel & Wolf, 1995). The concept of optical coherence is at third-generation synchrotron sources.

6 Singer and Vartanyants Coherence properties of focused X-ray beams J. Synchrotron Rad. (2014). 21,5–15 COHERENCE

longitudinal coherence Dl l

transverse coherence

d wc

z COHERENCE

longitudinal coherence λ2 lc ~ Dl Δλ l λ2 τc ~ cΔλ € Si (111) 0.5 um or 1.5 fs € transverse coherence λz w ~ c d d w c 34-ID-C 50m 25x70 um @ 9 keV

z € Courtesy Prof. Oleg Shpyrko (UCSD) Coherence: Laser Speckle

A. L. Schawlow “Laser Light" Scientific American, 219 (3), p. 120, (1968)

Laser Speckle: Interference pattern arising from randomly distributed scatterers Courtesy Prof. Oleg Shpyrko (UCSD)

SIMPLEST SPECKLE EXPERIMENT: TWINKLE, TWINKLE LITTLE STAR

Stars are big, but very far away. As a result, their light has a high transverse coherence. As the light propagates through the atmosphere our eye detects a portion of the coherent diffraction Courtesy Prof. Oleg Shpyrko (UCSD) First Speckle: First Speckle Photo: Exner, 1877 von Laue, 1914 (using candle light) (using arc discharge lamp)

K. Exner: Sitzungsber. Kaiserl. M. von Laue: Sitzungsber. Akad. Akad. Wiss. (Wien) 76, 522 (1877) Wiss. (Berlin) 44, 1144 (1914) Courtesy Prof. Oleg Shpyrko (UCSD) First X-ray Speckle: M. Sutton et al., Nature 352, 608-610 (1991)

2 I ∝ F(q) Courtesy Prof. Oleg Shpyrko (UCSD) WHY USE SYNCHROTRON RADIATION? Synchrotron sources offer:

• Brightnesss (small source, collimated) • Tunability (IR to hard x-rays) • Polarization (linear, circular) • Time structure (short pulses)

Source brightness is the key figure of merit for coherent imaging

B = photons/source area, divergence, bandwidth

2 Fc ~ l B COHERENT X-RAY REFERENCES

§ Vartanyants, I A, and A Singer. 2010 “Coherence Properties of Hard X-Ray Synchrotron Sources and X-Ray Free-Electron Lasers.” New Journal of Physics 12 (3): 035004. § Singer, Andrej, and Ivan A Vartanyants. 2014. “Coherence Properties of Focused X-Ray Beams at High-Brilliance Synchrotron Sources.” Journal of Synchrotron Radiation 21 (1). International Union of Crystallography: 5–15. doi:10.1038/nnano.2008.246. § Nugent, Keith. 2009. “Coherent Methods in the X-Ray Sciences.” Advances in Physics 59 (1): 1–99. doi:doi: 10.1080/00018730903270926.

10 Courtesy Prof. Oleg Shpyrko (UCSD)

X-RAYS, TWO EXTREMES FOR STRUCTURAL STUDY:

X-ray Imaging X-ray Diffraction

(Shadowgraphs) (Scattering) Coherent Imaging is scattering • Inhomogeneous, • Is particularly sensitive to non-periodic materials periodicity in the sample • Limited spatial resolution (Crystals) (~0.001 mm) • Atomic resolution (unit cell)

Von Laue, Nobel 1914 Roentgen, Nobel 1900 Bragg & Bragg, Nobel 1915 Courtesy Prof. Oleg Shpyrko (UCSD)

IMAGING REGIMES WITH COHERENT X-RAYS

near-field far-field Fraunhofer 2a Fresnel X-ray beam

z ~ a2/l z >> a2/l absorption phase in-line coherent radiograph contrast holography diffraction

Kagoshima (1999) Jacobsen (1990) Miao (1999) REFRACTIVE INDEX AND CONTRAST IN THE X-RAY REGION

re 2 n =1−δ − iβ =1− λ ni fi (0) A = A0 exp(−inkt) 2π ∑ i k = 2π /λ

RAPID COMMUNICATIONS

QUANTITATIVE X-RAY PHASE NANOTOMOGRAPHY PHYSICAL REVIEW B 85, 020104(R) (2012)

• Absorption contrast: (a) € (a) € (b) sensitive to Im(n) ~ 4πβ(x,y)t /λ

FIG. 3. (Color online) (a) Histogram of a region in the sample of approximately (2.8 µm)3 size marked with a square in Fig. 2(b).(b) • Phase contrast: The circles show the relative error of the histogram σr as a function of the voxel side length r, obtained by binning the voxels within this € region. The line shows a fit to the data (see text for details). sensitive to Re(n) (b) (c) (d) air 6 within a region in air σ 0.36 10− ,confirmingthatthe chosen area on the sphere= is homogeneous.× The knowledge ~ 2πδ(x,y)t /λ of the chemical composition of this material revealed a mass density of (1.05 0.05) g/cm3,whichagreedverywellwith the expected mass± density given by the manufacturer. The error of this measurement corresponds to a measurement on a single voxel, but binning the density in the selected volume reduces the spread of the distribution σ .Thiseffect Diaz,is shown Ana, in et Fig. al, 3(b)2012.,wheretherelativeerror “Quantitative X-Rayσr Phaseσ/δ0 € is plotted as a function of the voxel size. We observe= that Nanotomography.” Physical0.55 Review B 85. the error decreases as r− ,asindicatedbythefittedline. Assuming uncorrelated noise in the data, one would expect FIG. 4. (Color online) (a) 3D rendering of the ptychographic 1.5 afasterdecayofσ as r− .Theslowerdecayofthe tomogram of a sample consisting of 2-µm-diam SiO2 microspheres relative error in our measurement arises from correlated noise in a glass capillary. We show in yellow (dark gray) an isosurface introduced by the filtered back projection reconstruction21 representation of the microspheres, while the capillary is shown in a and from the ptychographic reconstructions. By choosing semitransparent light gray color. (b) 2D section through the tomogram abinningof323 voxels, corresponding to a voxel size of shown in (a). (c) 2D section of the same tomogram perpendicular approximately (1.4 µm)3,themeasureddensityonthatregion to the section shown in (b) at the position of the line. (d) Line is 1.048 0.008 g/cm3.Thisvalueactuallyprovidesavery profile indicated by a dashed line in (c) reveals the refractive index close match± with the value of the density of single polystyrene distribution within a particle and the capillary wall. particles reported by Grover et al.22 24 An analysis on a volume of the same size within the BaTiO3 limited amounts of sample are available. Even though the sphere revealed a histogram with a standard deviation σ recently developed techniques based on the nanomechanical 6 = 25 1.4 10− ,significantlylargerthantheonefoundinairor resonators used for the assessment of the densities of in the× polystyrene sphere. We attribute this to a lower degree single cells and single particles22 currently exhibit density of homogeneity within this sphere. Nevertheless, the average resolution superior to ptychographic tomography, they are density within this volume was 4.14 0.10 g/cm3 using a intrinsically limited to the provision of densities of the entire binning of 323 voxels, in agreement with± the value given by entities. Therefore, these techniques, unlike ptychographic the manufacturer. tomography, cannot provide the internal density maps, which Ptychographic tomography has further been applied to will prove very useful for the assessment of the phases of measure the mass density of individual microspheres. In multicomponent particles, such as core-shell colloidal spheres. order to illustrate this application, we show in Fig. 4(a) a In conclusion, we have shown that ptychographic tomogra- 3D tomogram of a sample of SiO2 microspheres of 2-µm phy yields accurate quantitative mass density values whenever 23 diameter confined in a SiO2 microcapillary. As expected, the specimen’s chemical composition is known. The sensitivity the tomogram revealed a homogeneous density within the of the density measurements has also been studied as a particles, especially visible in a line across one of the function of the binning of the 3D data, which can decrease microspheres, shown in Fig. 4(d).Aquantitativeanalysis statistical errors down to 1% in homogeneous samples. We over a 1-µm3 volume within this particle revealed a mass have further demonstrated the technique as a method to 3 density of (1.91 0.04) g/cm ,andsimilarvalueshavebeen measure the mass density of SiO2 microspheres with a relative obtained for the± other particles in the sample. The measured error of 2%. In this study the resolution is limited to 150 values were significantly smaller than expected for bulk SiO2. nm by mechanical stability and thermal drifts. However, the Silica microspheres can have significantly different densities resolution is fundamentally limited only by the wavelength than their corresponding bulk materials, and it is usually not and the detector angular extent. In practice, other limiting straightforward to determine the exact mass density when factors are an effective depth of focus related to the specimen’s

020104-3 REFRACTIVE INDEX AND CONTRAST IN THE X-RAY REGION re 2 A = A0 exp(−inkt) n =1−δ − iβ =1− λ ni fi (0) 2π ∑ k = 2π /λ i

€ €

A. Diaz,et al., Journal of Structural Biology, vol. 192, no. 3, pp. 461–469, Oct. 2015.

Absorption contrast: Phase contrast: sensitive to Im(n) sensitive to Re(n) ~ 4πβ(x,y)t /λ ~ 2πδ(x,y)t /λ

€ € A. I. FIGUEROA et al. PHYSICAL REVIEW B 90,174421(2014)

8 32 per Co atom obtained for a Co0.5Pt0.5 L10 alloy ﬁlm in Ref. [27]. In order to obtain mS/nh,onehastotakeintoaccount the contribution of the angular dependent magnetic dipole term 6 24 XAS Integrated mD since the Co0.5Pt0.5 L10 structure is highly anisotropic [27]. The angular dependence of mSeff is analogous to that of mL 4 16 deﬁned by Eq. (3). At the so-called “magic angle” (57.3◦), mD vanishes so that mSeff mS [27,56]. Thus, by applying the spin sum rule to the XMCD= signal obtained at the two measured 2 8

Normalized XAS Normalized angles (θ 0 and 60 )inourCo-Ptgranularﬁlms,we = ◦ ◦ have derived mSeff/nh(57.3 ) mS/nh 0.51(1)µB . Finally, ◦ = = 0 0 at θ 0◦mL/mSeff 0.11(1) and mL/mS 0.16(1), being = = = of the same order of that for a Co0.5Pt0.5 L10 alloy ﬁlm

XMCD Integrated 0 0 (mL/mS 0.15) [27]. XMCD= measurements at the Co L edges on Co Pt -2 -2 2,3 x 1 x particles prepared by vapor deposition have been recently− -4 -4 reported [57,58]. They find mL/mS ranging between 0.10 and 0.14 on particles of comparable size to ours, depending on the -6 -6 composition of the Co-Pt alloy present. Then, the mL/mS (for -8 -8 θ 0◦)ishigherinourCo-Ptsamples.Withreferencetoother Normalized XMCD Normalized XMCD= studies in Co-Pt systems, our m /m in Co is also -10 -10 L S larger, as compared to CoPt3 films deposited on a substrate 770 780 790 800 810 820 at 800 K (mL/mS 0.09) [26], and for annealed Co0.5Pt0.5 NPs with diameters= of about 2.6 nm deposited on amorphous carbon matrices (mL/mS 0.094) [10]. FIG. 17. (Color online) Co L2,3-edge spectra of the tCo 0.7nm = and t 1.5nmsample.(a)( )NormalizedXASspectramean= Pt = − curve measured at normal incidence. ( )Thedouble-stepfunction B. Pt L2,3-edge results used is also indicated. ( )Integratedwhiteline.(b)Angledependent The XMCD spectra measured at the Pt L2,3 edges are shown normalized XMCD spectra: ( ) normal incidence and ( )60◦. The in Fig.POLARIZED18.ThenonzerovaluesoftheXMCDsignalreflects X-RAYS GIVE SENSITIVITY TO integrated area of each XMCD signal is also plotted. theELECTRON polarization of the Pt by the SPIN magnetic Co.in From PtCo an XMCD experiment at the Pt L2,3 edges one obtains, through the sum the orbital magnetic moment, mL∥ and mL⊥ ,respectively,from the angle-dependent XMCD data. The orbital moment at each 1.6 measured angle mθ ,asobtainedfromthemagneto-opticalsum (a) L Pt - L rule [52], is expressed as 2,3 1.2 L2 θ 2 2 mL m∥ sin θ mL⊥ cos θ. (3) L = L + 3 0.8 Au -L However, in order to apply the magneto-optical sum 2,3 rules [52,53] to the XMCD signal in Fig. 17(b),thenumber 0.4 T=7K of unoccupied states in the 3d band for Co n is required in Normalized XAS h H=10kOe the calculations. Values of nh at the Co 3d band reported in the literature are, for example, nh 2.49 holes for metallic 0.0 = 0.05 (b) L 0.4 LCP @ 11.565keV Co [54]andnh 2.628 for Co0.5Pt0.5 L10 alloy films [27]. 2 =

Let us recall that the Co atoms in our Co-Pt granular samples integral XMCD 0.00 are found to be in these two different environments (metallic 0.0 Co and Co Pt L1 alloy), but we are not able to quantify 0.5 0.5 0 -0.05 p q -0.4 each of them. Therefore the right number of 3d holes in L3 Co atoms of our Co-Pt samples is unknown. Thus we have -0.10 -0.8 expressed the orbital moment as per 3d hole in the Co Normalized XMCD atoms. In addition, given the highly anisotropic character of -0.15 -1.2 the Co-Pt granular ﬁlm, the integrated 3d absorption white 11560 11600 13280 13320 RCP @ 11.565keV line r needs to be corrected for the angular dependence of the absorption intensity, which we have done following the Photon energy (eV) Figueroa, A. I., et al. (2014). Physical Review B, 90(17), 174421 procedure of Ref. [55]. We obtain m∥ /nh 0.05(1) µB and L = FIG. 18. (Color online) ( )NormalizedXANES(a)and() mL⊥ /nh 0.08(1) µB from the application of the orbital sum XMCD (b) at the Pt L edges in the Co-Pt sample with t 0.7 = 2,3 Co = rule to the XMCD at the two incident angles (θ 0◦ and 60◦). nm and t 1.5 nm. ( ) XMCD integrated area. (--)TheXANES = Pt = From the spin sum rule [53], we derive an effective spin spectrum of Au L2,3 edges is shown in (a) for comparison. The higher magnetic moment per 3d hole of mSeff/nh 0.77(1) µB , at limit for the p and q integrals, used for the sum rules analysis, is given = θ 0 .ThisvalueiscomparabletothemSeff/nh 0.75 µB in (b). = ◦ = 174421-12 CDI IN BRAGG GEOMETRY: IMAGING DISPLACEMENT FIELD (STRAIN)

Displacement field u(r): G

No phase structure

Coherent X-ray Diffraction measures:

Strain is a gradient of u(r), the phase component of the complex-valued density TRADITIONAL Courtesy Prof. Oleg Shpyrko (UCSD) MICROSCOPY:

Object Plane Detector Plane Lens-less Imaging: Courtesy Prof. Oleg Shpyrko (UCSD)

FIRST DEMONSTRATION OF CDI WITH X-RAYS

diffraction pattern reconstruction

J. Miao, Nature 400, 342 (1999) COHERENT DIFFRACTIVE IMAGING: Y. Takahashi et al., Phys. Rev. B 82, 214102 (2010) LARGE-FORMAT, SINGLE-PHOTON SENSITIVE X-RAY CCD CAMERAS OPENED THE DOOR TO COHERENT X-RAY IMAGING

Fairchild Peregrine 486 CCD Camera

• 4K x 4K pixel array (61.4 mm square area) • 15 μm pixels, 100% fill factor • Back-illuminated for up to 80% QE • Readout noise < 5 e- at 50 Kpixels/s • Dynamic range > 86 dB in MPP • 6 s readout with four on-chip amplifiers • Pixel binning for more rapid readout • Peltier-cooled to -50 C for low dark current Hi Resolution Imaging (CCD)? 210nm

At APS 34-ID-C: ~7nm data 9.25 hours of scanning 38 minutes of x-ray exposure

Rainbow color map (still) considered harmful http://ieeexplore.ieee.org/document/4118486/21 PIXEL ARRAY DETECTORS: REVOLUTIONIZING COHERENT IMAGING

D. Schuette, S. Gruner (Cornell)

PADs can be read out in ~1 ms (CCDs take seconds!)

Pilatus 6M detector PAD pixels are 55-150 µm. (PSI/Dectris) (CCDs are 12-24 µm) Hi Resolution Imaging (PAD)?

Quad Timepix GaAs sensor 700nm gold crystal 3 degree rocking curve 25 minute measurement 15 sec movie (1X APS-U measurement) Coherent Diffraction from Crystals

|Fourier Transform|2

24 Coherent Diffraction from Crystals

|Fourier Transform|2

25 Measuring 3D CXD Silver Nano Cube (111)

CCD

ki Q=kf-ki

Yugang Sun and Younan Xia, Science 298 2177 (2003)

26 3D Ag Nano Cube

Yugang Sun and Younan Xia, Science 298 2177 (2003)

200nm

27 APS 34-ID-C

Precision scanning stage

28 ORIGINAL PHASE RETRIEVAL PAPER (BY D. SAYRE):

NOT r=r2 Acta Cryst. (1952). 5, 60 Critical sampling Object N unknown densities 1.1 1 N/2 equations 0.9 0.8 0.7 0.6 & 0.5 *+",- 0.4 !" = $ '() 0.3 % 0.2 0.1 0 -0.1 Nyquist Freq = 1/(2*bandwidth) -5 45 95 145 195 245 Intensity is under sampled

FFT Real Part Intensity 5 0.1

4.5 0.09

4 0.08 3.5 0.07 3 0.06 2.5 0.05 2 0.04 1.5 0.03 1 0.5 0.02

0 0.01

-0.5 0 100 110 120 130 140 150 160 30 100 110 120 130 140 150 160 Oversampling 2x Object N/2 equations 1.1 1 0.9 N/2 Unknown densities 0.8 0.7 0.6 N/2 Known Densities(zero) 0.5 0.4 0.3 & 0.2 *+",- 0.1 !" = $ '() 0 % -0.1 0 50 100 150 200 250

Oversampled FFT Intensity 8 5

7 4.5

6 4

5 3.5

4 3

3 2.5

2 2

1 1.5

0 1

-1 0.5

-2 0 100 110 120 130 140 150 160 100 110 120 130 140 150 160 Input Output Algorithms

FFT

Reciprocal Space Direct Space Constraints Constraints

Experimental Amplitudes Support Oversampled Total Energy Thresholded Mixing previous iterates. & Positivity ZeroPadded

FFT -1 J. R. Fienup Appl. Opt. 21 2758 (1982) Collins Nature 298, 49 (1982) R. W. Gerchberg and W. O. Saxton Optik 35 237 (1972) Fienup, James R. 2013. “Phase Retrieval Algorithms: a Personal32 Tour.” Applied Optics 52 (1): 45–56. Input Output Algorithms

FFT t(n) Reciprocal Space Direct Space Constraints Constraints

Experimental Amplitudes Support Oversampled Total Energy Thresholded Mixing old and new iterates. & Positivity ZeroPadded u(n) FFT − 1 J. R. Fienup Appl. Opt. 21 2758 (1982) Collins Nature 298, 49 (1982) R. W. Gerchberg and W. O. Saxton Optik 35 237 (1972) Fienup, James R. 2013. “Phase Retrieval Algorithms: a Personal33 Tour.” Applied Optics 52 (1): 45–56. Input Output Algorithms

FFT t(n)

Reciprocal Space Error Reduction Direct Space Constraints % (n) Constraints (n ) τ i i ∈ Support ui = & ' 0.0 i ∉ Support Experimental Amplitudes Support Oversampled Total Energy Thresholded Hybrid Input-Output Mixing old and new iterates. € (n ) & ' τ i ∈ Support Positivity u(n ) = ( i ZeroPadded i (n−1) (n ) ) ui - βτ i i ∉ Support

(n) € u FFT − 1 J. R. Fienup Appl. Opt. 21 2758 (1982) Collins Nature 298, 49 (1982) R. W. Gerchberg and W. O. Saxton Optik 35 237 (1972) Fienup, James R. 2013. “Phase Retrieval Algorithms: a Personal34 Tour.” Applied Optics 52 (1): 45–56. Input Output Algorithms

FFT t(n) Reciprocal Space Direct Space Constraints Constraints Hybrid Input-Output Experimental Amplitudes Support + τ (n ) i ∈ Support ∩ ϕ ≤ ϕ Oversampled (n ) i i max Total Energy ui = , (n−1) (n ) Thresholded - ui - βτ i i ∉ Support ∪ ϕi > ϕmax Mixing old and new iterates. & Positivity ZeroPadded € u(n) FFT − 1 Harder, R., Liang, M., Sun, Y., Xia, Y., & Robinson, I. K. (2010). Imaging of complex density in silver nanocubes by coherent x-ray diffraction. New Journal of Physics, 12(3), 035019. Huang, X., Harder, R., Xiong, G., Shi, X., & Robinson, I. (2011). Propagation uniqueness in three- dimensional coherent diffractive imaging. Physical Review B, 83(22), 224109. 35 Input Output Algorithms

FFT t(n) Reciprocal Space Direct Space Constraints Constraints

Experimental Amplitudes Support Oversampled Total Energy Thresholded Mixing previous iterates. & Positivity Zero Padded

u(n) FFT − 1 Harder, R., Liang, M., Sun, Y., Xia, Y., & Robinson, I. K. (2010). Imaging of complex density in silver nanocubes by coherent x-ray diffraction. New Journal of Physics, 12(3), 035019. Huang, X., Harder, R., Xiong, G., Shi, X., & Robinson, I. (2011). Propagation uniqueness in three- dimensional coherent diffractive imaging. Physical Review B, 83(22), 224109. 36 Monitor Reciprocal Space Error

23nm

37 GENETIC/GUIDED ALGORITHM

Random Random Random Random Random Random Random start start start start start start start

HIO/ER HIO/ER HIO/ER HIO/ER HIO/ER HIO/ER HIO/ER

Fitness= Fitness= Fitness= Fitness= Fitness= Fitness= Fitness= 0.01 0.3 0.8 0.05 0.7 0.65 0.5

Fitness metrics Cross Reciprocal Space Error Cross the Sqrt(a*b) Sharpness (sum pho^4) population with the best. … …

Individual Individual Individual Individual Individual Individual Individual 1 2 3 4 5 6 7

Clark, J. N., Ihli, J., Schenk, A. S., Kim, Y.-Y., Kulak, A. N., Campbell, J. M., et al. (2015). Three-dimensional imaging of dislocation propagation during crystal growth and dissolution. Nature Materials, 14(8), 780–784. http://doi.org/10.1038/nmat4320

Clark, J. N., Huang, X., Harder, R., & Robinson, I. K. (2012). High-resolution three-dimensional partially coherent diffraction imaging. Nature Communications, 3, 993–. http://doi.org/10.1038/ncomms1994 38 ROSS J. HARDER (XSD) MATHEW J. CHERUKARA (XSD) YOUSSEF NASHED (MCS) TOM PETERKA (MCS) PRASANNA BALAPRAKASH (MCS) S. SANKARANARAYANAN (CNM) BADRI NARAYANAN (MSD)

drhgfdjhngngfmhgmghmghjmghfmf TRAINING THE NEURAL NETWORK - NN Reciprocal Real space space intensity Encoder1 Decoder1 intensity sBCDI Training: 180,000 examples generated 20,000 held back for validation 1,000 used for testing Deep neural networks

- NN Reciprocal Real space space intensity Encoder2 Decoder2 phase BCDI p

Cherukara, M. J., Nashed, Y. S. G., & Harder, R. J. (2018). Scientific Reports, 8(1), 16520. http://doi.org/10.1038/s41598-018-34525-1

40 NEURAL NETWORK THAT LEARNS IMAGE RECONSTRUCTION

A Input intensity B True object C Predicted object D True phase E Predicted phase 3D Ag Nano Cube

Yugang Sun and Younan Xia, Science 298 2177 (2003)

200nm

42 3D Strain Map in ZnO

2µm

43 Nature Materials 9, 120 - 124 (2010) 3D Strain Map in ZnO

44 Nature Materials 9, 120 - 124 (2010) 3D Strain Map in ZnO

45 Nature Materials 9, 120 - 124 (2010) Patterned Gold Nanocystal Samples

46 Multiple reflection reconstructions

(11-1)

1-11

-111