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Coherent Diffraction Imaging 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 à 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 à 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 ! theÂà 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 ; ! Pà 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.
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