Ptychography

Ptychography (/t(a)ɪˈkɒgrəfi/ t(a)i-KO-graf-ee) is a computational microscopy technique that reconstructs the complex-valued image (amplitude and phase) of a specimen from a series of coherent diffraction patterns recorded as a localized probe is scanned with overlap across the sample. It unifies the principles of microscopy and crystallography, combining the real-space imaging of microscopy with the reciprocal-space diffraction analysis of crystallography to produce high-resolution, quantitative images free from lens aberrations. Ptychography has been demonstrated with visible light, X-rays, electrons and extreme-ultraviolet radiation, enabling quantitative phase contrast imaging across nine orders of magnitude in length scales. Its defining characteristic is translational invariance, which means that the interference patterns are generated by one constant function (e.g. a field of illumination or an aperture stop) moving laterally by a known amount with respect to another constant function (the specimen itself or a wave field). The interference patterns occur some distance away from these two components, so that the scattered waves spread out and "fold" (Ancient Greek: πτυχή, "ptychē" is 'fold') into one another as shown in the figure. Unlike conventional lens imaging, ptychography is unaffected by lens-induced aberrations or diffraction effects caused by limited numerical aperture. This is particularly important for atomic-scale wavelength imaging, where it is difficult and expensive to make good-quality lenses with high numerical aperture. Another advantage is its high phase sensitivity, enabling clear imaging of transparent or weakly absorbing specimens. This is because it is sensitive to the phase of the radiation that has passed through a specimen, and so it does not rely on the object absorbing radiation. In the case of visible-light biological microscopy, this means that cells do not need to be stained or labelled to create contrast. Modern ptychography, developed in the 2000s and now the most widely used form of the technique, combines scanning microscopy with coherent diffractive imaging (CDI) through iterative phase-retrieval algorithms. In this approach, a coherent probe—such as an X-ray, electron, or optical beam—is scanned across the specimen with overlapping illumination regions, and a diffraction pattern is recorded at each position. The overlap between adjacent probe positions in real space and diffraction data in reciprocal space provide sufficient redundancy to enable the simultaneous reconstruction of both the probe and the sample transmission functions. This yields quantitative, aberration-free phase images that are robust to partial coherence and experimental imperfections, while providing both high spatial resolution and a large field of view. Modern ptychography has been demonstrated with X-rays, electrons, and visible light, providing sub-ångström resolution in electron microscopy and quantitative three-dimensional imaging through X-ray ptychotomography.

Phase recovery

Although the interference patterns used in ptychography can only be measured in intensity, the mathematical constraints provided by the translational invariance of the two functions (illumination and object), together with the known shifts between them, means that the phase of the wavefield can be recovered by an inverse computation. Ptychography thus provides a general solution to the "phase problem". By recording more independent intensity measurements than unknown pixels in object function (due to the overlap constraint in real space), the phase information can be solved for computationally using iterative algorithms. This formulation has also stimulated substantial research in applied mathematics, particularly on the uniqueness, stability, and convergence properties of phase-retrieval problems. Once this is achieved, all the information relating to the scattered wave (modulus and phase) has been recovered, and so virtually perfect images of the object can be obtained. There are various strategies for performing this inverse phase-retrieval calculation, including direct Wigner distribution deconvolution (WDD) and iterative methods. The difference map algorithm developed by Thibault and co-workers is available in a downloadable package called PtyPy.

Optical configurations There are many optical configurations for ptychography: mathematically, it requires two invariant functions that move across one another while an interference pattern generated by the product of the two functions is measured. The interference pattern can be a diffraction pattern, a Fresnel diffraction pattern or, in the case of Fourier ptychography, an image. The "ptycho" convolution in a Fourier ptychographic image derived from the impulse response function of the lens.

The single aperture

This is conceptually the simplest ptychographical arrangement. The detector can either be a long way from the object (i.e. in the Fraunhofer diffraction plane), or closer by, in the Fresnel regime. An advantage of the Fresnel regime is that there is no longer a very high-intensity beam at the centre of the diffraction pattern, which can otherwise saturate the detector pixels there.

Focused-probe ptychography

A lens is used to form a tight crossover of the illuminating beam at the plane of the specimen. The configuration is used in the scanning transmission electron microscope (STEM), and often in high-resolution X-ray ptychography. The specimen is sometimes shifted up or downstream of the probe crossover so as to allow the size of the patch of illumination to be increased, thus requiring fewer diffraction patterns to scan a wide field of view.

Multislice ptychography

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