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Ptychographic Fresnel Coherent Diffractive Imaging

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Ptychographic Fresnel Coherent Diffractive Imaging 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 re- quire that the reconstruction obey the ptychographic con- straint that adjacent overlapping regions of the complex exit- *[email protected] surface wave field be consistent ͓13͔. The advantages of 1050-2947/2009/80͑6͒/063823͑5͒ 063823-1 ©2009 The American Physical Society VINE et al. PHYSICAL REVIEW A 80, 063823 ͑2009͒ 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 radia- tion. 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, 063823-2 PTYCHOGRAPHIC FRESNEL COHERENT DIFFRACTIVE … PHYSICAL REVIEW A 80, 063823 ͑2009͒ ␺m͑ ͒ ␺ ͑ ͒ ␳m͑ ͒ ͑ ͒ ͑focal length f =13.8 mm͒.
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