Ptychography
Ptychography is a computational method of microscopic imaging. It generates images by processing many coherent interference patterns that have been scattered from an object of interest. Its defining characteristic is translational invariance, which means the interference patterns are generated by one constant function moving laterally by a known amount with respect to another constant function. The interference patterns occur some distance away from these two components, so that the scattered waves spread out and ‘fold’ into one another as shown in the figure.
Ptychography can be used with visible-light, X-rays, extreme ultraviolet or electrons. 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 important advantage of the technique is that it allows transparent objects to be seen very clearly. 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.
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, 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 comprehensive solution to the so-called ‘phase problem’. Once this is achieved, all the information relating to the scattered wave 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 and iterative methods. The Difference Map developed by Thibault and co-workers is available as a downloadable package called .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, 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.Focussed 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, 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.Near-field ptychography
This uses a wide field of illumination. To provide magnification, a diverging beam is incident on the specimen. An out-of-focus image, which appears as a Fresnel interference pattern, is projected onto the detector. The illumination must have phase distortions in it, often provided by a diffuser that scrambles the phase of the incident wave before it reaches the specimen, otherwise the image remains constant as the specimen is moved, so there is no new ptychographical information from one position to the next. In the electron microscope, a lens can be used to map the magnified Fresnel image onto the detector.Fourier ptychography
A conventional microscope is used with a relatively small numerical aperture objective lens. The specimen is illuminated from a series of different angles. Parallel beams coming out of the specimen are brought to a focus in the back focal plane of the objective lens, which is therefore a Fraunhofer diffraction pattern of the specimen exit wave. Tilting the illumination has the effect of shifting the diffraction pattern across the objective aperture. Now the standard ptychographical shift invariance principle applies, except the diffraction pattern is acting as the object and the back focal plane stop is acting like the illumination function in conventional ptychography. The image is in the Fraunhofer diffraction plane of these two functions, just like in conventional ptychography. The only difference is that the method reconstructs the diffraction pattern, which is much wider than the aperture stop limitation. A final Fourier transform must be undertaken to produce the high-resolution image. All the reconstruction algorithms used in conventional ptychography apply to Fourier ptychography, and indeed nearly all the diverse extensions of conventional ptychography have been used in Fourier ptychography.Imaging ptychography
A lens is used to make a conventional image. An aperture in the image plane acts equivalently to the illumination in conventional ptychography, while the image corresponds to the specimen. The detector lies in the Fraunhofer or Fresnel diffraction plane downstream of the image and aperture.Bragg ptychography or reflection ptychography
This geometry can be used either to map surface features or to measure strain in crystalline specimens. Shifts in the specimen surface, or the atomic Bragg planes perpendicular to the surface, appear in the phase of the ptychographic image.Vectorial ptychography
Vectorial ptychography needs to be invoked when the multiplicative model of the interaction between the probe and the specimen cannot described by scalar quantities. This happens typically when polarized light probes an anisotropic specimen, and when this interaction modifies the state of polarization of light. In that case, the interaction needs to be described by the Jones formalism, where field and object are described by a two-component complex vector and a 2×2 complex matrix, respectively. The optical configuration for vectorial ptychography is similar to those of classical ptychography, although a control of light polarization needs to be implemented in the setup. Jones maps of the specimens can be retrieved, allowing to quantify a wide range of optical properties. Similarly to scalar ptychography, the probes used for the measurement can be jointly estimated together with the specimen.Advantages
Lens insensitive
Ptychography can be undertaken without using any lenses at all, although most implementations use a lens of some type, if only to condense radiation onto the specimen. The detector can measure high-angles of scatter, which do not need to pass through a lens. The resolution is therefore only limited by the maximum angle of scatter that reaches the detector, and so avoids the effects of diffraction broadening due to a lens of small numerical aperture or aberrations within the lens. This is key in X-ray, electron and EUV ptychography, where conventional lenses are difficult and expensive to make.Image phase
Ptychography solves for the phase induced by the real part of the refractive index of the specimen, as well as absorption. This is crucial for seeing transparent specimens that do not have significant natural absorption contrast, for example biological cells, thin high-resolution electron microscopy specimens, and almost all materials at hard X-ray wavelengths. In the latter case, the phase signal is also ideal for high-resolution X-ray ptychographic tomography. The strength and contrast of the phase signal also means that far fewer photon or electron counts are needed to make an image: this is very important in electron ptychography where damage to the specimen is a major issue that must be avoided at all costs.Tolerance to incoherence
Unlike holography, ptychography uses the object itself as an interferometer. It does not require a reference beam. Although holography can solve the image phase problem, it is very difficult to implement in the electron microscope where the reference beam is extremely sensitive to magnetic interference or other sources of instability. This is why ptychography is not limited by the conventional ‘information limit’ in conventional electron imaging. Furthermore, ptychographical data is sufficiently diverse to remove the effects of partial coherence that would otherwise affect the reconstructed image.Self-calibration
The ptychographical data set can be posed as a blind deconvolution problem. It has sufficient diversity to solve for both the moving functions, which appear symmetrically in the mathematics of the inversion process. This is now routinely done in any ptychographical experiment, even if the illumination optics have been previously well characterised. Diversity can also be used to solve retrospectively for errors in the offsets of the two functions, blurring in the scan, detector faults, like missing pixels, etc.Inversion of multiple scattering
In conventional imaging, multiple scattering in a thick sample can seriously complicate, or even entirely invalidate, simple interpretation of an image. This is especially true in electron imaging. Conversely, ptychography generates estimates of hundreds or thousands of exit waves, each of which contains different scattering information. This can be used to retrospectively remove multiple scattering effects.Robustness to noise
The number counts required for a ptychography experiment is the same as for a conventional image, even though the counts are distributed over very many diffraction patterns. This is because dose fractionation applies to ptychography. Maximum likelihood methods can be employed to reduce the effects of Poisson noise.Applications
Applications of ptychography are diverse because it can be used with any type of radiation that can be prepared as a quasi-monochromatic propagating wave.Ptychographic imaging, along with advances in detectors and computing, has resulted in the development of X-ray microscopes. Coherent beams are required in order to obtain 'far-field' diffraction patterns with speckle patterns. Coherent X-ray beams can be produced by modern synchrotron radiation sources, free-electron lasers and high harmonic sources. In terms of routine analysis, X-ray ptycho-tomography is today the most commonly used technique. It has been applied to many materials problems including, for example, the study of paint, imaging battery chemistry, imaging stacked layers of tandem solar cell, and the dynamics of fracture. In the X-ray regime, ptychography has also been used to obtain a 3-D mapping of the disordered structure in the white Cyphochilus, and a 2-D imaging of the domain structure in a bulk heterojunction for polymer solar cell.
Visible light ptychography has been used for imaging live biological cells and studying their growth, reproduction and motility. In its vectorial version, it can also be used for mapping quantitative optical properties of anisotropic materials such as biominerals.
Electron ptychography is uniquely sensitive to both heavy and light atoms simultaneously. It has been used, for example, in the study of nano-structure drug delivery mechanisms by looking at drug molecules stained by heavy atoms within light carbon nanotubes cages. With electron beams, shorter wavelength, higher-energy electrons used for higher-resolution imaging can cause damage to the sample by ionising it and breaking bonds, but electron-beam ptychography has now produced record-breaking images of molybdenum disulphide with a resolution of 0.039 nm using a lower-energy electron beam and detectors which are able to detect single electrons, so atoms can be located with more precision.
Ptychography has several applications in the semiconductor industry, including imaging their surfaces using EUV, their 3D bulk structure using X-rays, and mapping strain fields via Bragg ptychography, for example in nanowires.