ARTICLE IN PRESS Ultramicroscopy 109 (2009) 338–343 Contents lists available at ScienceDirect Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic Probe retrieval in ptychographic coherent diffractive imaging Pierre Thibault a,Ã, Martin Dierolf a,b, Oliver Bunk a, Andreas Menzel a, Franz Pfeiffer a,b a Paul Scherrer Institut, 5232 Villigen PSI, Switzerland b E´cole Polytechnique Fe´de´rale de Lausanne, 1015 Lausanne, Switzerland article info abstract Article history: Ptychography is a coherent diffractive imaging method that uses multiple diffraction patterns obtained Received 25 September 2008 through the scan of a localized illumination on the specimen. Until recently, reconstruction algorithms Received in revised form for ptychographic datasets needed the a priori knowledge of the incident illumination. A new 4 December 2008 reconstruction procedure that retrieves both the specimen’s image and the illumination profile was Accepted 23 December 2008 recently demonstrated with hard X-ray data. We present here the algorithm in greater details and illustrate its practical applicability with a visible light dataset. Improvements in the quality of the PACS: reconstruction are shown and compared to previous reconstruction techniques. Implications for future 42.30.Rx applications with other types of radiation are discussed. 42.30.Va & 2009 Elsevier B.V. All rights reserved. 42.30.Kq 42.15.Dp Keywords: Phase retrieval Ptychography Diffractive imaging Difference map algorithm 1. Introduction edges—a situation documented as hard, if not impossible to treat in the traditional reconstruction schemes [5]. In certain situations, Most coherent diffractive imaging (CDI) methods rely on far- the problem can still be solved if the wavefield incident on the field diffraction data to retrieve the complex-valued projection or specimen is also well known. This so-called ‘‘key-hole CDI’’ was three-dimensional density of a sample. Unless the sample is demonstrated recently [6,7]. periodic, as is the case in crystallography, finite coherence length Data analysis can be further eased if redundancy in the data is and fixed dimensions of the detector pixels impose a limit on the introduced through multiple measurements with different but amount of information carried by the wave for the problem to be partly overlapping illuminated regions. This concept, called tractable. For this reason, all CDI techniques require ‘‘oversam- ‘‘ptychography’’, was introduced nearly 40 years ago for applica- pling’’ of the diffraction pattern, i.e. that the region in real-space tions in electron microscopy [8]. Ptychography was recently contributing to the diffraction data does not extend beyond a limit reformulated to be used with a simpler and efficient iterative predefined by the experimental conditions. In many experiments, reconstruction algorithm [9] whose potential was demonstrated oversampling is achieved with a plane wave incident on an lately with visible light [10] and X-rays [11]. isolated specimen. Despite the successes of this approach [1–4], Using hard X-rays, it was recently demonstrated that both the the requirement for sufficiently small and isolated samples is an specimen’s transmission function and the illumination profile can important obstacle to a wider application of the technique. be extracted from a ptychographic dataset [12]. In this article, we To overcome the isolated specimen limitation, one has to describe in greater details the reconstruction approach, and depart from the special case of planar incident waves, and rather demonstrate its possibility with a laser light experiment. Our select a region of the specimen with the illumination itself, either algorithm is based on the difference map, a general algorithm that with a mask or a focused beam. This experimental convenience solves constraint-based problems. It differs from another recently can however lead to insurmountable reconstruction difficulties. published method [13] that also addresses the probe retrieval Propagation effects produce illuminated regions with smooth problem. After describing the algorithm and addressing the question of uniqueness of the solution, we will present the reconstruction results and discuss the implications of the ability à Corresponding author. Tel.: +4156 310 3705. to retrieve the illumination profile for future applications of E-mail address: [email protected] (P. Thibault). ptychography. 0304-3991/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2008.12.011 ARTICLE IN PRESS P. Thibault et al. / Ultramicroscopy 109 (2009) 338–343 339 2. Reconstruction algorithm individual view: P ðCÞ : c ! cF ¼ p ðc Þ. (4) Detailed descriptions of ptychography can be found in a F j j F j number of earlier publications [14,15]. A specimen is illuminated Given an estimate cj of the jth view, pFðcjÞ is obtainedpffiffiffi by with a finite and coherent incident wave called the ‘‘probe’’. The replacing the magnitudes of the Fourier transform by Ij while wave emerging from the specimen propagates to the far field, keeping the original phases. This operation, not always called a where a pixel-array detector measures its intensity. A complete projection, is ubiquitous in iterative phase retrieval methods in dataset consists of a set of such diffraction patterns, each imaging and crystallography (see for instance Ref. [22]). corresponding to a known displacement of the probe relative to The overlap projection can be computed from the minimiza- the specimen. Hence, one has tion of the distance kC À COk2, subject to the constraint (3). The ^ ^ 2 calculation entails to finding O and P that minimize IjðqÞ¼jF½Pðr À rjÞOðrÞj , (1) X X O 2 ^ ^ 2 kC À C k ¼ jcjðrÞPðr À rjÞOðrÞj . (5) where Ij is the jth measured diffraction pattern, P is the probe j r function, O is the specimen’s transmission function (the object) and F denotes the Fourier transform operation, with the The associated projection is reciprocal space coordinate q. Reconstruction of the image O ^ ^ POðCÞ : c ! c ðrÞ¼Pðr À rjÞOðrÞ. (6) amounts to recovering a function O that satisfies (1) for all j’s. j j Early solutions of ptychographic problems where obtained Minimization of Eq. (5) must be carried numerically as closed with a method called ‘‘Wigner deconvolution’’ [14]. Simultaneous form expressions for O^ and P^ cannot be obtained. For this task, reconstruction of the object and the probe functions was various methods such as conjugate gradients minimization can be attempted in this formalism [16], in which it becomes a blind used efficiently. In particular, line minimization of (5) involves deconvolution problem. Although very elegant, Wigner deconvo- only finding the root of a polynomial of order 3. Alternatively, O 2 ^ ^ lution requires that the probe positions rj lie on a grid as fine as setting to zero the derivative of kC À C k with respect to P and O the aimed resolution, making it impractical in many situations. gives the solution as a system of equations P More recently, Rodenburg and Faulkner [9] have introduced an ^ à P ðr À rjÞc ðrÞ iterative reconstruction algorithm inspired from other commonly O^ ðrÞ¼ Pj j , (7) ^ 2 used phase retrieval algorithms, such as Fienup’s hybrid inpu- jjPðr À rjÞj P t–output [17]. Called ‘‘ptychographic iterative engine’’ (PIE), the ^ à jO ðr þ rjÞcjðr þ rjÞ algorithm cycles through all datasets sequentially to update the P^ðrÞ¼ P . (8) jO^ ðr þ r Þj2 object function. The algorithm bears some resemblance with a j j general form described by Youla [18] and Fla˚ m [19] for convex set In the event the probe P^ is already known, the overlap feasibility problems, every step being viewed as an under-relaxed projection is given by (6), where O^ is computed with Eq. (7). If P^ projection. While its formulation is natural and intuitive, it is also needs to be retrieved, both Eqs. (7) and (8) need to be unclear how PIE can be generalized to allow the simultaneous simultaneously solved. While the system cannot be decoupled reconstruction of the probe with the object. analytically, applying the two equations in turns for a few The reconstruction procedure described in this paper departs iterations was observed to be an efficient procedure to find the from both previously known methods. It uses the difference minimum. Within the reconstruction scheme, initial guesses for P^ map [20], an iterative algorithm originally devised for phase and O^ are readily available from the previous iteration—apart retrieval but now known to have a much wider range of from the very first iteration for which a rough model of the probe applications [21]. The algorithm is designed to solve problems is needed to start the search for a minimum. that can be expressed as the search for the intersection point The reconstruction is implemented with the above projections, between two constraint sets. Incidentally, the ptychographic using the difference map. With a standard choice of parameters problem (1) can be re-expressed trivially as two intersecting [20], the iterative procedure takes a form similar to Fienup’s constraints by introducing the exit waves fcjg (the ‘‘views’’ on the hybrid input–output algorithm [17], specimen) corresponding to each position of the probe on C ¼ C þ P ½2P ðC ÞC ÀP ðC Þ. (9) the specimen. The Fourier constraint expresses compliance to nþ1 n F O n n O n the measured intensities, Owing to the algorithm’s very wide range of applications, it is doubtful that a general proof of convergence exists, as is the case 2 Ij ¼jFcjj , (2) for instance in the narrower case of projections on convex sets [18,23]. General properties of the algorithm have been studied in and the overlap constraint states that each view can be factorized detail by Elser [24] and its converging behavior has been verified as a probe and an object function: in multiple situations. cjðrÞ¼Pðr À rjÞOðrÞ. (3) The initial state vector is typically generated with a random number generator, although transient time can be reduced with a This reformulation is the crucial step in the design of our better initial estimate of the views.