High-Resolution Scanning X-ray Diffraction Microscopy

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Science  18 Jul 2008:
Vol. 321, Issue 5887, pp. 379-382
DOI: 10.1126/science.1158573


Coherent diffractive imaging (CDI) and scanning transmission x-ray microscopy (STXM) are two popular microscopy techniques that have evolved quite independently. CDI promises to reach resolutions below 10 nanometers, but the reconstruction procedures put stringent requirements on data quality and sample preparation. In contrast, STXM features straightforward data analysis, but its resolution is limited by the spot size on the specimen. We demonstrate a ptychographic imaging method that bridges the gap between CDI and STXM by measuring complete diffraction patterns at each point of a STXM scan. The high penetration power of x-rays in combination with the high spatial resolution will allow investigation of a wide range of complex mesoscopic life and material science specimens, such as embedded semiconductor devices or cellular networks.

Imaging techniques that rely on coherence, loosely designated as coherent diffractive imaging (CDI) techniques, have thrived in situations where more traditional modes of microscopy are difficult to implement. CDI allows high-resolution imaging while eliminating the need to use high-quality lenses, especially hard to produce in the case of x-rays. This experimental convenience comes, however, at the price of harder data analysis, which often entails reconstructing the image by solving the so-called phase problem. The simplest CDI method, called diffraction microscopy, involves the reconstruction of an image or a three-dimensional (3D) density from far-field diffraction intensities. In the past decade, a number of successful demonstrations of this technique have been reported, ranging from fabricated nanostructures in 2D (1, 2) and 3D (3, 4) to biological specimens in 2D (5, 6). Requirements such as the complete isolation of the specimen have promoted the development of alternative approaches, such as Fourier holography (7, 8) and “ptychography” (913). The latter is an experimental method developed in the 1970s for electron microscopy (14), which consists of measuring multiple diffraction patterns by scanning a finite illumination on an extended specimen. Overlap between adjacent illumination positions provides overdetermination in the data. Assuming the profile of the illumination is known, solving the phase problem has been demonstrated to be easier.

Another powerful microscopy technique is scanning transmission x-ray microscopy (STXM). It consists in scanning a focused x-ray beam on a specimen and measuring the transmitted intensity at each raster point. It thus provides a map of the specimen's transmission. Although more advanced imaging modes exist (15, 16), common to all of them is that the resolution is limited by the spot size on the specimen. Increase in resolution thus largely depends on improving focusing optics. STXM relies partly on coherence—higher coherence allows smaller spot sizes—but has developed quite independently from CDI methods.

In this work, we have designed a technique that we call scanning x-ray diffraction microscopy (SXDM), which bridges the gap between scanning microscopy and coherent diffractive methods. To demonstrate this method experimentally, we imaged a buried nanostructure by using a focused hard x-ray beam. The apparatus, described below, is similar to the experimental setups used in both fields. Data acquisition was greatly facilitated by the use of Pilatus, a fast single-photon counting detector having no read-out noise (17). In essence, the experiment follows closely the configuration in which ptychography was first developed. We used a separate reconstruction method (18) to complement the standard STXM analysis tools and reached resolutions about five times higher than the focal spot dimensions. Our algorithmic approach overcomes the main difficulty of ptychographic measurements by automatically recovering the incident illumination profile at the same time as the complex-valued image of the specimen. This feature is critical to obtain high-quality reconstructions because the detailed structure of the wavefield incident on the specimen is difficult to extract by other means.

A schematic view of the experimental setup is shown in Fig. 1A. An incoming coherent x-ray beam is focused down to a few hundred nanometers on a specimen. The intense focal spot, called the probe, interacts with a small portion of the specimen before propagating out to a pixel array detector in the far-field region. One complete data set is produced by recording diffraction patterns as the specimen is scanned through the focal spot, typically on a 2D grid.

Fig. 1.

Schematic representation of the experimental setup. (A) The incoming hard x-ray beam is focused on the specimen. A pixel array detector records a complete oversampled diffraction pattern for each raster position of the specimen. (B) Examples of diffraction patterns. The color scale is logarithmic, and the units are in photon counts. The focused beam geometry generates a diverging beam past the specimen, thereby removing the need for a beam stop at the detector.

Each measured diffraction pattern carries information on the specimen. More precisely, the wave at the exit of the specimen corresponding to the jth diffraction data set can be expressed as Embedded Image(1) where P(r-rj) is the probe wavefield translated by a known amount rj, and O(r) is the specimen's transmission function, called the object (19). Exit waves ψj are called views because they effectively provide information on selected regions of the specimen. Redundancy in the data, needed for ptychographic applications, is enforced with a scanning step size small enough for adjacent views to overlap.

To demonstrate the technique, we measured a SXDM data set, using as a specimen a Fresnel zone plate buried under a gold layer (20). Imaging this type of nanofabricated device demonstrates the penetration power of x-rays and allows a reliable assessment of the resolution and the validity of a reconstruction. The complete data set is a raster scan of 201-by-201 diffraction patterns, produced with a 6.8-keV x-ray beam focused to a 300-nm spot size, each exposed for 50 ms (18). Four of these diffraction patterns are shown in Fig. 1B. These diffaction patterns were sampled on a region of the detector that subtended an angular range of 10 mrad by 10 mrad.

As immediate feedback to the microscopist to define regions of interest, for instance, the SXDM data set can be analyzed by using standard scanning microscopy techniques (15, 16). A simple calculation of the zeroth and first moments of the intensity distribution in each diffraction pattern gives low-resolution images of the transmission and of the gradient of the phase shift produced as the wave traverses the specimen (21).

The result of this STXM analysis on a SXDM data set is shown in Fig. 2, B to D. Whereas the spot size of 300 nm limits the resolution of these preliminary micrographs, it takes on quite a different role in the following analysis, namely, to ensure that the diffraction pattern is sufficiently “oversampled” by the detector in the far field.

Fig. 2.

Preliminary analysis of a measured data set of 201-by-201 diffraction patterns. This test specimen is a Fresnel zone plate of 30-μm diameter and with 70-nm outer zone width. (A) SEM image. The zone plate structure is completely buried under a layer of gold. A full reconstruction of the region of interest (solid line) is shown in Fig. 3. The reconstructed image of the dashed line region is shown in fig S1. (B) Transmission of the specimen, obtained from the total transmitted intensity at each probe position. (C) Phase gradient of the transmitted wave in the horizontal direction. (D) Phase gradient in the vertical direction. (B) to (D) involve no phase retrieval and can be produced as data are acquired. The pixel size is equal to the scanning grid spacing (100 nm), but the resolution is limited by the size of the probe (about 300 nm).

The approach we have adopted to reconstruct a high-resolution image from a SXDM data set departs from both earlier methods, that is, the Wigner deconvolution (9, 11) and the ptychographic iterative engine (22, 23). We used the difference map (24), an iterative algorithm now known to have a wide range of applications going far beyond imaging (25). Problems solved by the difference map are formulated as a search for the intersection between two constraint sets. Any solution attempt is represented by a state vector, which typically satisfies neither constraint until the solution is found. For phase retrieval, the state vector generally is a 2D image or a 3D density. For SXDM reconstructions, it is given by the collection of views corresponding to each position of the probe relative to the object. The two constraints on the state vector are the Fourier and the overlap constraints. The former enforces consistency of the views with their corresponding diffraction patterns (26). The latter is given by Eq. 1 and formalizes the redundancy of information in the data set, which gives this algorithm, like related ptychographic techniques (9, 11, 22, 23), its robustness. Projections onto both constraint sets can be precisely defined and easily implemented in a computer program.

At each iteration, the current state is updated through a combination of the two projections, as prescribed by the difference map algorithm. Each computation of the overlap projection yields new estimates of both the probe and the object (27). Unlike many diffraction microscopy reconstructions, this type of problem typically features very short convergence times, a few tens of iterations. It should also be noted that SXDM reconstructions do not suffer from the defocus ambiguity that is met in other CDI methods. The multiplicative relation between the probe and the object ensures that the plane of the reconstruction is well defined.

A reconstruction example, from a 61-by-61 subset of the 201-by-201 diffraction patterns data set, is given in Fig. 3. A second region of interest, closer to the center of the zone plate, is shown in fig. S1. The retrieved complex-valued image provides a map of the optical transfer function of the specimen. The amplitude component carries the absorption information, and the phase expresses the phase shift experienced by a wave when traversing the specimen. In the present case, the phase difference caused by the 1.0-μm-thick gold structure in the zones is measured to be about 0.67 π, in very good agreement with the expected value of 0.65 π. We note that a complex reconstruction carries the necessary information to reproduce any traditional microscopy arrangement. For instance, a differential phase contrast image, similar to Fig. 2, C and D, is simply given by the gradient of Fig. 3B. An example of a simulated phase contrast image is shown in fig. S1C.

Fig. 3.

SXDM reconstruction of the complex optical transmission function of the zone plate specimen. (A) Amplitude and (B) phase of the reconstruction of a selected region of 61-by-61 diffraction patterns, from the same data set that was used for Fig. 2. The magnified region shows imperfections in the nanofabricated zones. Other defects, corroborated by a SEM image, are shown in fig. S1. The outermost rings of the zone plate (70 nm wide) are very well resolved. The pixel size is 18 nm. The total acquisition time for this image is 186 s, with a total dose of 2 × 109 photons.

The wavefield of the probe, which is retrieved at the same time as the object, is shown in Fig. 4A. The knowledge of both the phase and the amplitude of the wave gives the most complete information for subsequent analysis. Computed propagation of this wavefield shows that the specimen was placed 1.2 mm behind the focal plane and was thus illuminated with a slightly curved and diverging wavefront.

Fig. 4.

Reconstructed probe. (A) Color rendition of the complex-valued probe at the specimen plane. (B) The computed wavefield propagated back to the focal plane, situated 1.2 mm upstream. (C) A section of the probe wavefield along the propagation direction, as calculated from the data in (A). The dashed line, indicating the plane of (B), is at the waist of the wave distribution. (D) The squared amplitude of the Fourier transform of the probe. (E) Measured diffraction pattern of the probe without a specimen. This data, which was not used in the reconstruction, is in good agreement with the retrieved probe.

Unlike the few other known methods (11, 28), the probe reconstruction does not require any knowledge of the pupil of the Fresnel zone plate. In fact, it does not require a pupil plane. Probes formed with refractive (29) or reflective (30) optics can be equally well reconstructed, as well as unfocused probes formed with a mask. The door to a general wavefront characterization procedure appears to be even more widely open when one realizes the symmetry in the multiplicative relation between P and O in Eq. 1. Transferring the oversampling requirement to the object function placed downstream—for instance, by using a simple pinhole—allows the characterization of arbitrary large and extended regions of an incoming wavefield. In this wavefront-sensing configuration, what was the “probe” becomes the main object of interest: It can be the propagated wave perturbation from any specimen or optical device. This feature will be useful for the characterization of focusing devices to be used at future coherent x-ray light sources.

We have demonstrated an imaging technique that is fully compatible with STXM and has the high-resolution potential of CDI approaches. The reconstruction method removes the main difficulty of ptychography by retrieving the complex-valued image of both the specimen and the probe. The method is noninvasive, and radiation damage can be reduced by a combination of fast large-area scans with longer high-resolution imaging of regions of interest. The technique can be used with hard and soft x-rays, can be combined with other scanning methods such as fluorescence imaging, and can be extended to nanodiffraction mapping or nanospectroscopy. Higher coherent flux and improved focusing optics (31) should soon provide conditions for better-than-10-nm resolutions, making possible the imaging of the finest structures in state-of-the-art electronics devices or the macromolecular assemblies in organic tissues. Future developments comprise extension of the method to 3D reconstructions of inorganic and organic specimens as a possible answer to the current limitations of diffraction microscopy (32).

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