# Ptychography

Ptychography /ti-CHO-graf-e/ is a technique invented by Walter Hoppe[1] that aims to solve the diffraction-pattern phase problem by interfering adjacent Bragg reflections coherently and thereby determine their relative phase. In the original formulation, Hoppe envisaged that such interference could be effected by placing a very narrow aperture in the plane of the specimen so that each reciprocal-lattice point would be spread out and thus overlap with one another. The name ptychography, from the Greek for fold, derives from this optical configuration; each reciprocal lattice point is convolved with some function, and thus made to interfere with its neighbors. In fact, measuring only the intensities of interfering adjacent diffracted beams still leads to an ambiguity of two possible complex conjugates for each underlying complex diffraction amplitude. The original formulation of ptychography is equivalent to the well known theorem that for a finite specimen (that is one delineated by a narrow aperture, sometimes known as a finite support), the one-dimensional phase problem is soluble to within an ambiguity of 2N, where N is the number of Fourier components that make up the specimen.[2] However, such ambiguities may be resolved by changing the phase, profile or position of the illuminating beam in some way.[1] The fact that not only the intensities of the diffracted beams but also the intensities lying midway between the beams, where the convolved Bragg beams interfere, is an alternative statement of the Nyquist–Shannon sampling theorem for components of diffracted intensity. These components generally have twice the frequency (in reciprocal space) of their underlying complex amplitudes.

Ptychographic imaging along with advances in detectors and computing have resulted in X-ray microscopes, optical and electron microscopy with increased spatial resolution without the need for lenses.[3][4]

The use of X-rays demands a high degree of spatial coherence in order to obtain far-field diffraction patterns with speckle patterns. This property is necessary in order to perform coherent diffraction imaging (CDI) and ptychographic experiments. Coherent X-ray beams could be produced by modern synchrotron radiation sources and free electron lasers.

In diffraction experiments, only intensities of scattered radiation can be recorded. All phase information is lost. The phase problem can also be solved in this case with the application of certain constraints. However, with the single exposure diffraction pattern, there is a limitation on the size of the investigated sample: it should be less than the transverse size of the coherent part of the beam. Also it is also not trivial to make the reconstruction converge to a stable solution.

The idea of X-ray ptychography is to scan an extended sample through the beam while adjacent probe spots overlap significantly in the object plane. This produces a redundancy of the diffracted data which could be inverted with ptychographical iterative engine algorithm (PIE).[5] Its main advantages are noise tolerance, high speed of convergence, and possible high resolution of reconstructed images.

Recently, ptychography, which was initially developed in a scalar formalism, has been generalized to a vectorial formalism.[6] This improvement is relevant in cases where the specimen presents anisotropic features so that the interaction with light depends on its state of polarization. This is especially the case in the optical regime, where many artificial or natural materials present birefringent properties. By using a proper set of polarized probes and polarized detection scheme, vectorial ptychography allows the retrieval of the birefringent and/or diattenuation properties of the specimen.