These routines are based on propagation of the optical field between detector and object plane, calculated by Fourier transforms. by using one of the iterative phase retrieval routines. CDI allows a complete recovery of a non-periodic object from its far-field diffraction pattern, provided the latter is sampled with at least twice the Nyquist frequency (oversampling), 13 13.ĭ. which is applied to single-particle diffraction patterns recorded at free electron laser facilities. To demonstrate our method we select the most popular modern coherent imaging techniques, CDI, 10 10. Here an iterative routine is applied, which includes the following steps: (i) The second step consists of extrapolation of U 0. Thus, obtaining the complex-valued distribution of U 0 constitutes the first step of our method. These elementary waves can be extrapolated well beyond the detector of size S 0 × S 0, and thus effectively increase the numerical aperture and hence the resolution. The key of our method is that the distribution U 0 is complex-valued and thus contains sufficient information to uniquely define the elementary waves scattered by the object. The back-propagation of the wave U 0 to the object domain results in the reconstruction of the object at a resolution provided by the Abbe criterion R 0 = λ/(2N.A. The complex-valued wave U 0 forming the interference pattern I 0 can be reconstructed by employing conventional numerical methods. In a typical experiment, a finite fraction of an interference pattern I 0, such as a hologram or coherent diffraction pattern, is recorded by a detector of size S 0 × S 0, and digitized with N 0 × N 0 pixels, so that S 0 = N 0Δ, where Δ is the pixel size of the detector. Here, we propose a universal approach for post-extrapolation of experimental coherent interference patterns that allows extrapolation and resolution enhancement even without phase information available from an experimental record. However, in a general case of coherent imaging, a reference wave is not provided and only the amplitude of the complex-valued scattered wave can be captured, thus the method 3 3. The ingenious way of providing such phase information is holography, where the unknown object wave is superimposed with a well-known reference wave. Previously, it has been reported that provided the complex-valued scattered wavefront, in particularly its phase, is known, it can be extrapolated beyond the size of the recorded interference pattern increasing the resolution of the reconstructed object. However, this very interference term contains the phase information about the interfering waves, and, in the technique we propose here, it allows reconstructing the entire complex-valued wavefront created by the scatterers. Now, the interference term U 1 U 2* + U 1* U 2 obscures the image of two scatterers. For example, for two point scatterers, the total intensity in case of incoherent radiation is given by I = | U 1| 2 + | U 2| 2, where U 1 and U 2 are complex-valued waves diffracted by scatterers 1 and 2 while in case of coherent radiation, the total intensity is given by I = | U 1| 2 + | U 2| 2 + U 1 U 2* + U 1* U 2. Coherent radiation, despite many obvious advantages, deteriorates the resolution due to interference effects between the scattered waves. However, with the invention of optical lasers, and later, coherent X-ray and electron sources, imaging techniques employing coherent waves have been developed, and here the Abbe's criterion is only remotely related to the possibly achievable resolution. it has been the quantitative measure of optical system performance until today. Ever since Ernst Karl Abbe introduced the term “Numerical Aperture” (N.A.) and proposed the resolution criterion R = λ/(2N.A.), 1,2 1.Į.
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