Ptychographic Fresnel Coherent Diffractive Imaging

Home , Fraunhofer diffraction, Fresnel diffraction

PHYSICAL REVIEW A 80, 063823 ͑2009͒

Ptychographic Fresnel coherent diffractive imaging

D. J. Vine,1,* G. J. Williams,1 B. Abbey,1,2 M. A. Pfeifer,3 J. N. Clark,3 M. D. de Jonge,4 I. McNulty,5 A. G. Peele,3 and K. A. Nugent1 1School of Physics, The University of Melbourne, Parkville, Victoria 3010, Australia 2Department of Engineering and Science, University of Oxford, Oxford OX1 2JD, United Kingdom 3Department of Physics, La Trobe University, Bundoora, Victoria 3086, Australia 4Australian Synchrotron, Clayton, Victoria 3168, Australia 5Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 6043, USA ͑Received 4 September 2009; published 8 December 2009͒ This paper reports improved reconstruction of complex wave ﬁelds from extended objects. The combination of ptychography with Fresnel diffractive imaging results in better reconstructions with fewer iterations required to convergence than either method considered separately. The method is applied to retrieve the projected thickness of a gold microstructure and comparative results using ptychography and Fresnel diffractive imaging are presented.

DOI: 10.1103/PhysRevA.80.063823 PACS number͑s͒: 42.30.Rx, 42.30.Wb

I. INTRODUCTION Fresnel diffraction pattern being measured in the far field of the object, and is known as Fresnel CDI ͑FCDI͓͒9͔. The Imaging with x-ray radiation allows the internal structure physical properties of the lens do not limit the achievable of optically opaque materials to be investigated nondestruc- resolution in FCDI. In general, we consider monochromatic, tively and at a higher resolution than is possible with visible spatially coherent incident radiation; however, considerations light. Direct imaging methods are limited by the resolution due to partial coherence have also been investigated ͓10͔. of x-ray lenses which has driven the development of an in- In addition to overcoming some of the limitations of CDI direct, lensless method known as coherent diffractive imag- with plane wave illumination, the use of a divergent incident ing ͑CDI͓͒1–6͔. beam enables imaging of nonisolated samples ͑i.e., which do In CDI, the action of the image-forming lens is replaced not have finite support͒, if the incident beam is sufficiently with a post-measurement computation step. The lens how- localized so that it defines the support region ͓11,12͔. This ever is sensitive to the complex wave-field incident upon it means the technique can be used to scan regions of an ex- and we are able to measure the squared modulus of the field tended object, a capability which has enormous practical im- only. To duplicate the lens computationally we must first plications for increasing the scope of CDI. recover the phase associated with the measured wave-field On the other hand, Fresnel diffraction is not invariant with ͑ modulus. While the phase problem imposes some computa- respect to translation of the diffracting object i.e., sample ͒ tional difficulty which must be overcome, the advantages of drift during measurements , a property possessed of CDI that diffractive imaging include wavelength limited resolution results from the Fourier shift theorem. The practical implica- and quantitative-amplitude and -phase information. tion is that FCDI experiments can be affected by sample stability issues. In the simplest variant of CDI, a plane wave illuminates Ptychography ͓13–19͔ is a related iterative technique in an isolated object and a detector in the far field of the object which the exit-surface wave of an object lacking finite records the squared modulus of the diffraction pattern. The support can be reconstructed from the modulus of its Fraun- wave fields at the exit-surface plane of the object and at the hofer diffraction pattern. In ptychography, plane-wave radia- detector are linked via a Fourier transform and the complex ͓ ͔ tion is incident on an aperture and the sample placed a small exit-surface wave can be recovered iteratively 7,8 by im- distance downstream. The radiation probe is defined by the posing a priori known constraints in the object and detector pinhole and stepped across the object, with significant planes. Typically, these are the detector-plane modulus ͑posi- ͑Ն ͒ ͓ ͔ ͒ 60% overlap between adjacent positions 20 , and a dif- tive square root of the measured intensity and a finite sup- fraction pattern recorded in the far field of the object for each port in the sample plane which obeys the sampling require- scan point. The reconstruction procedure in ptychography ment. This simple implementation of CDI has limitations imposes, in addition to the constraints imposed by CDI, the such as slow numerical convergence, stagnation prior to con- requirement that overlapping reconstructions of the complex verging and nonuniqueness of the solution with respect to wave field in the sample plane be mutually consistent. diffraction homologs ͑distinct objects sharing a common ͒ In this paper, we show how the techniques of FCDI and Fraunhofer diffraction pattern modulus . These limitations ptychography may be fused into a hybrid technique for dif- can be overcome by illuminating the sample with a divergent fractive imaging, ptychographic Fresnel CDI ͑P-FCDI͒.In beam such as that emerging from the focal plane of a Fresnel ͓ ͔ ͑ ͒ P-FCDI, we extend the keyhole CDI method 11 to illumi- zone plate FZP . Such illuminating radiation results in a nating overlapping positions on the sample and further require that the reconstruction obey the ptychographic constraint that adjacent overlapping regions of the complex exit- *[email protected] surface wave field be consistent ͓13͔. The advantages of

FIG. 2. ͑Color online͒ Ptychographic FCDI is implemented by stepping the sample, in this case a gold microstructure in the shape of an “X,” through the beam and recording a diffraction pattern for each sample position. These diffraction patterns will be used to reconstruct the exit-surface complex wave ﬁeld of the sample.

gold microstucture and soft x rays. Using these results we compare P-FCDI to FCDI and divergent beam ptychography.

II. ANALYSIS The P-FCDI reconstruction algorithm used to analyze the experimental data is illustrated in Fig. 1. Figure 2 illustrates FIG. 1. The ptychographic Fresnel CDI algorithm. The flow- a plane wave is focused to a point upstream of the sample. chart shows that the FCDI algorithm forms an inner loop of con- The focus/sample distance determines the size of the beam ventional ptychography. on the sample and the phase curvature of the incident radiation. A central stop on the FZP and order-sorting aperture ͑ ͒ implementing reconstructions in this way are: i improve- located at the focal plane selects the first-order diffraction of ment in the reconstruction at the edges of the beam support the FZP. compared to FCDI alone; and ͑ii͒ incorporation of the ben- ␺ The focused radiation I—assumed to be a known efits of FCDI into ptychography. In addition, the reconstruc- quantity—partially illuminates the sample with an unknown tion of the extended object in P-FCDI is recognizable at the transmission function, O, and gives rise to an exit-surface conclusion of the first ptychographic iteration. wave field, ␺ which can be written under the projection In plane-wave CDI and ptychography the diffraction pat- i tern in general bears no resemblance to the object being il- approximation as luminated. In P-FCDI an image of the sample may be recog- ␺m͑ ͒ ␺ ͑ ͒ ͑ ͒ ͑ ͒ i r,0 = I r,0 O r − ri , 1 nizable in the detected diffraction pattern to the extent that alignment is greatly simplified. Furthermore, a detector where r=͑x,y͒ is a two-dimensional position vector in the beamstop is unnecessary. This efficient use of all photons plane perpendicular to the direction of x-ray propagation, ri scattered from the sample is carried over into the P-FCDI describes the discrete lateral translations of the sample in the method. beam to give the i-th scan position and m specifies the cur- We note that the use of a divergent beam means our rent ͑FCDI͒ iteration. method is ptychography inspired but not strictly ptychogra- We define the modulus constraint operator M˜ and support phy since that method typically uses an aperture to define the constraint operator Sˆ as radiation probe. The benefits of FCDI derive from having ␺͑q͒ significant phase curvature over the illuminating probe radia- M˜ ͓␺͑q͔͒ = ͉␺ ͑q͉͒ , ͑2͒ tion and this is why the divergent beam from an FZP is used. M ͉␺͑q͉͒ The use of a pinhole in ptychography to define the probe radiation does not have, in the limit of small sample-pinhole Sˆ͓␺͑r͔͒ = ⍀͑r͒␺͑r͒͑3͒ distances, a significant phase curvature and thus does not ͉␺ ͉ enjoy the advantages of FCDI. The scanning x-ray diffrac- where q is the Fourier space dual to r, M is the known tion microscopy technique ͑SXDM͓͒18͔ also uses a lens to modulus and ⍀ is typically a binary function equal to unity form the probe radiation. However, in SXDM the sample sits inside the supported region and zero elsewhere. ␺ at the focal plane of the lens. Henceforth we use the term The known quantity I has been predetermined using the ptychography to refer to divergent wave illumination. method of ͓21͔ and we use this to decompose the wave field ␺m In the Sections that follow we introduce the formalism of i into a coherent sum of the illuminating and scattered field P-FCDI and apply it to experimental data obtained with a components, ␳ ͓9͔,

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␺m͑ ͒ ␺ ͑ ͒ ␳m͑ ͒ ͑ ͒ ͑focal length f =13.8 mm͒. The sample and detector were i r,0 = I r,0 + i r,0 , 4 placed a distance of 620 ␮m and 530 mm from the focal ␳m͑ ͒ ͓ ͑ ͒ ͔␺ ͑ ͒ and i r,0 = Om r−ri −1 I r,0 . Crucially, the function plane respectively. The detector was a direct detection CCD ␳ ͑ has a good approximation to finite support defined by the with 2048ϫ2048 13.5 ␮m pixels. ͒ ␺ beam when the incident radiation I is sufficiently well lo- The sample was a commercially available binary gold mi- ͓ ͔ calized 9 . A measure of this localization is given by the crostructure in the shape of an ‘X’ about 10 ␮m in diameter ratio of the area of the supported region to the total area. In ͑Xradia resolution and calibration standard X-40–30–2͒. The this experiment we chose the support by threshholding the test pattern had a thickness of 150Ϯ15 nm and was intensity at five percent of the maximum, an empirically de- mounted on a silicon nitride window of approximately 110 termined value, giving a ratio of less than one percent and nm thickness. we conclude that the localization approximation is justified. Four beam positions were chosen to scan the sample with Propagation of the wave field between the sample ͑z=0͒ and detector planes ͑z=z͒ proceeds via the Fresnel propaga- an overlap of between 20–40 % of the support. The sample was then scanned through the four beam positions and 500 tor ͓22͔ expressed here in operator format ͓23͔, Dˆ , z exposures of 1 s duration were recorded at each position. Images were summed to improve the signal to noise however ␺m͑q,z͒ = Dˆ ͓␺m͑r,0͔͒. ͑5͒ i z i this was complicated by sample drift over the course of the The ͑m+1͒th estimate of the scattered wave field is given by experiment. To overcome this limitation only a susbset of the the operator equation, measured images that were sufficiently well correlated were used in the analysis ͓9͔. For our data we calculated the cross ␳m+1͑ ͒ ˆ͓ ˆ −1͕ ˜ ͓␳m͑ ͒ ␺ ͑ ͔͒ ␺ ͑ ͖͔͒ ͑ ͒ ͑ ͒ i r,0 = S Dz M i q,z + I q,z − I q,z , 6 correlation of one arbitrarily chosen image from the data set with all others and then used the largest contiguous block ␺ ͑ ͒ ˆ ͓␺ ͑ ͔͒ ␳0͑ ͒ where I r,z =Dz I r,0 and the initial estimate i r,0 which was above a minimum correlation coefficient of 0.97. is Typically only 30–100 exposures were used from those recorded at each beam position. The resolution of the recon- ␳0͑ ͒ ˆ −1͓͉␺ ͑ ͉͒ ␸͑ ͒ ␺ ͑ ͔͒ ͑ ͒ i r,0 = Dz M q exp i q − I q,z 7 structed wave field, ϳ50 nm, was limited by the signal-to- noise ratio of the measured diffraction patterns. We used the ␸͑ ͒ for some arbitrary real function q , typically random num- a priori knowledge that the object was composed of a single bers or a constant. After a predetermined number of itera- material with known ratio of the refractive index decrement tions on ␳m ͑e.g., m=M͒ an estimate of the the object trans- to the attenuation coefficient to expedite numerical conver- mission function O is updated according to gence. ͉␺ ͑ ͉͒ In the following we compare reconstructions of the gold I I r − ri,0 On+1͑r͒ = On͑r͒ + ␤͚ microstructure using three different analysis techniques: i=1 max͕͉␺ ͑r,0͉͖͒ I FCDI, ptychography and P-FCDI. The FCDI reconstruction ϫ͓␺M ͑ ͒ ␺M ͑ ͔͒ ͑ ͒ n+1,i r,0 − n,i r,0 . 8 technique considers each illumination position on the sample separately ͑no feedback between ovelapping positions͒ and ␤ The parameter is an empirically determined variable that the final image is a composite image of the individual recon- controls the degree of feedback in the algorithm. Variation in structions. The ptychography reconstruction algorithm is es- ␤ allows more emphasis to be placed on the current or pre- sentially the same as P-FCDI with the important difference vious estimates of the reconstructed wave field. The first that there is only ever one consecutive iteration between term in the sum of Eq. ͑8͒ ensures that the object function is sample and detector planes before moving to the next sample updated in regions where the signal is largest. We use an additional physical constraint to define the position. The P-FCDI reconstruction method is extensively plane of reconstruction to be the sample’s exit surface. Under detailed in the preceding section. the projection approximation, the wave field at the sample’s Figure 3 compares the reconstructed projected thickness exit-surface plane must ͑i͒ be phase advanced relative to of the sample using P-FCDI, ptychography and FCDI. All ͑ vacuum, and ͑ii͒ have a positive attenuation coefficient. In reconstructions used the same parameters except where the absence of this constraint the reconstructed wave field specified͒ to facilitate cross comparison. A total of 800 itera- may show Fresnel diffraction effects, and must then be back- tions were used for each reconstruction. This was split be- propagated to the sample plane. tween the different reconstruction techniques as follows FCDI; 4:200:1 ͑beam positions:FCDI iterations:ptychography iterations͒, P-FCDI; 4:10:20, and ptychography; 4:1:200. III. EXPERIMENTAL METHOD The composite FCDI reconstruction was created by averag- The experiment was conducted at the 2-ID-B beamline of ing the projected thicknesses. the Advanced Photon Source ͑Argonne National Laboratory, It can be seen from comparing Figs. 3͑a͒ and 3͑c͒ that one Argonne IL, USA͓͒24͔. A 2.15 keV ͑␭=5.76 Å͒ beam of x advantage of combining FCDI and ptychography yields an rays was selected by a multilayer spherical grating mono- improvement in the uniformity of the complex wave field at chromator and made to be incident on an FZP with a diam- the boundary between adjacent beam positions. The P-FCDI eter of 160 ␮m and an outermost zone width of 50 nm algorithm resulted in more uniform thickness reconstructions

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(a) (b) 160

120

80 thickness (nm) 40

0 (c) 160 (d)

120

thickness (nm) 40

0 0 1234 distance (µm)

FIG. 3. ͑Color online͒ Comparison of reconstructions of the projected thickness of the gold microstructure ͑a͒ reconstruction using FCDI, ͑b͒ ptychography, and ͑c͒ using P-FCDI. ͑d͒ Line profiles through the projected thickness with the shaded region representing the expected thickness from the manufacturers specification. The line weightings of the scale bars in ͑a͒, ͑b͒ and ͑c͒ provide the key to the line profiles in ͑d͒ and indicate a scale of 8 ␮m. compared to ptychography and we attribute this refinement lows the complex exit-surface wave field of extended objects to the effect of the previously mentioned constraint which lacking finite support to be reconstructed from their diffrac- specifies the plane of the reconstruction within the depth of tion patterns. The resulting composite reconstruction shows focus. Since ptychography does not iterate multiple times pronounced improvement at the boundary of the illuminating between sample and detector planes this constraint was ob- beam compared to single image FCDI and compares favor- served to have less effect. ably to a conventional ptychographic reconstruction in re- ͑ ͒ The discrepancy in Fig. 3 d between the reconstructed moving reconstruction artifacts, but with a substantially and expected projected thickness may be due to the assump- smaller overlap requirement, fast iterative convergence, and tion that the sample is bulk density gold. If the gold was rapid sample alignment. We anticipate that this technique can deposited at less than bulk density the calculated projected be profitably employed to investigate a wide range of mate- thickness from the reconstructed phase should be greater. rials and biological systems. All of the reconstructions show some concentric circular artifacts centered on the each beam position which is symp- tomatic of FCDI phase retrieval. These artifacts are likely due to fluctuations in the illuminating radiation over the ACKNOWLEDGMENTS course of the experiment which was conducted over 24 h. ␺ Any method which using a single value for I will contain such artifacts as the illumination varies over the course of the The authors acknowledge the financial support of the Aus- measurement. tralian Research Council Centre of Excellence for Coherent X-ray Science. Use of the Advanced Photon Source at Ar- IV. CONCLUSION gonne National Laboratory was supported by the U.S. De- We have presented a method of diffractive imaging which partment of Energy, Office of Science, Office of Basic En- combines ptychography with Fresnel CDI. Our method al- ergy Sciences under Contract No. DE-AC02-06CH11357.

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