Lensless coherent imaging by a deterministic phase retrieval method with an aperture-array filter

Abstract

Recently we have proposed a noniterative and analytic phase retrieval method using the filter of an aperture array to reconstruct a complex-valued object from a diffraction intensity pattern [Phys. Rev. Lett.98, 223901 (2007)], but this method suffers from the restriction of the far-field condition of two distances between the object and the filter and between the filter and the detector for the intensity measurement. An improved method, which extends the adaptable condition of those distances to the region of Fresnel diffraction, is proposed here. In addition, the procedure for reducing the influence of noises on the phase retrieval is presented, in which the phase information contained in multiple groups of sampling data of a single intensity distribution is utilized. The usefulness of this method is shown in computer-simulated examples of the object reconstructions, including an object with phase vortices.

Coherent diffractive imaging beyond the Fresnel approximation using a deterministic phase-retrieval method with an aperture-array filter

“…As described in the previous paper [13], the diffraction pattern of each aperture in the detector plane is isolated approximately from those of the adjoining square if the parameters of the measurement system (i.e., z, l, w and d) satisfy a condition d1 l∕z > ε2λl∕w lσ j ∕z, (j u, v), where d1 l∕z denotes the array's period in the detector plane given from the geometrical optics prediction for the wave propagation from the small object, 2λl∕w lσ j ∕z(in which σ j is the extent of the object function in the direction of u or v) represents the main width of the diffraction pattern of each square aperture in the case of the far-field diffraction, and ε indicates a safety factor larger than unity for increase of the main width due to the phase variation of the wavefront inside each aperture. Note that this is not a necessary condition but a sufficient one.…”

“…In this section, a summary of the object reconstruction method by use of the array filter [12,13] is described for the sake of readability. Figure 1 shows a schematic diagram of the optical system used in the experiment as described in the following section.…”

“…As described in the previous paper [13], the present phase-retrieval method requires that the inverse Fourier transform of the aperture function R 0 x; y has an extent large enough to be approximated into a Gaussian function within the extent of the object. This Gaussian approximation is satisfied when the Fresnel number N F concerning the aperture function in Eq.…”

“…However, the distance z 35.44 mm and l 44.51 mm meet the requirement that the Fresnel number in Eq. (6) is less than unity (i.e., N F ≅ 0.200 for the parameters), which is needed to utilize a Gaussian approximation of the square aperture's function in the phase retrieval [13].…”

“…Object imaging from a single-shot diffraction intensity measurement is especially important among observational systems in the various approaches, because it enables us to observe the dynamic behavior of an object, and also it allows for the reduction of sample damage by an x-ray or electron-wave illumination. Recently, we have proposed a novel method of coherent diffractive imaging using a deterministic phaseretrieval method with an aperture-array filter [12,13], which allows for the reconstruction of a complex-valued object from a single diffraction intensity measured with the filter. This method does not utilize a reference wave or an iterative algorithm with an object's support constraint.…”