The Phase Problem
There are two relevant parameters for diffracted waves: amplitude and phase. In typical microscopy using lenses there is no phase problem, as phase information is retained when waves are refracted. When a diffraction pattern is collected, the data is described in terms of absolute counts of photons or electrons, a measurement which describes amplitudes but loses phase information. This results in an ill-posed inverse problem as any phase could be assigned to the amplitudes prior to an inverse Fourier transform to real space.
Three ideas developed that enabled the reconstruction of real space images from diffraction patterns. The first idea was the realization by Sayre in 1952 that Bragg diffraction under-samples diffracted intensity relative to Shannon’s theorem. If the diffraction pattern is sampled at twice the Nyquist frequency (inverse of sample size) or lower it can yield a unique real space image. The second was an increase in computing power in the 1980s which enabled iterative Hybrid input output (HIO) algorithm for phase retrieval to optimize and extract phase information using adequately sampled intensity data with feedback. This method was introduced by Fienup in the 1980s. Finally, the development of “phase recovery” algorithms led to the first demonstration of CDI in 1999 by Miao.
Read more about this topic: Coherent Diffraction Imaging
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