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.

A noniterative method for retrieving the phase of a wave field from a diffraction intensity measurement in a coherent wave field is proposed. In this method, the phase can be calculated from analytic equations based on the properties of entire functions by use of Fourier transforms. This method requires only a single measurement of the intensity of a diffracted wave through an array filter of rectangular apertures and then does not need to use any lens systems and coherent reference waves. Therefore, it provides a potentially useful means for coherent imaging in a wide class of wave fields such as optical, x ray, electron, and atomic waves.

Recently, we have proposed a coherent diffractive imaging using a noniterative phase retrieval method with the filter of an aperture array. The first (to our knowledge) experimental demonstration of this coherent imaging is presented here, in which a complex-valued object illuminated by a diode laser is reconstructed from the isolated diffraction intensities of the object's wave field, transmitted through an array filter of square apertures by using the phase retrieval method. This imaging method requires only a single measurement of the diffraction intensity and does not need a tight object's support constraint utilized in iterative phase retrieval algorithms or a reference wave used in holographic techniques.

An improved method for resolution of object reconstruction using phase retrieval by use of a scanning slit aperture is proposed. The reconstruction is based on measurements of the Fraunhofer diffraction intensities of wave fields transmitted through a scanning slit in the Fresnel-zone plane of an object. In the improved method, the measurement coordinates of the intensities depend not only on the slit's position used in a previous method but also on the slit's position scaled by the ratio between two distances among the object, Fresnel-zone, and detector planes. The spatial-frequency band for the object reconstruction, which is limited in a previous method by the extent of the Fourier transform of the slit function, can be extended to the bandwidth dependent on the scanning area with the slit. In addition, even in the measurement of the Fresnel diffraction intensities of wave fields transmitted through the slit, the improved resolution can be retained by compensation for a transverse shift of the intensities.

Almost all noninterferometric phase-retrieval methods used in coherent diffractive imaging have been based on the measurement system with low numerical aperture, in which Fresnel or Fraunhofer approximation is valid to express the wave propagation between an object and a detector. In microscopy, which is a typical application of coherent diffractive imaging, the measurement of the diffraction intensity with high numerical aperture is required for object reconstruction at high spatial resolution. We here propose an extension procedure to apply the previous phase-retrieval method using an aperture-array filter [J. Opt. Soc. Am. A25, 742 (2008)] to the system with high numerical aperture, in which the first Rayleigh-Sommerfeld integral for spherical waves is utilized instead of the Fresnel integral for parabolic waves. Computer-simulated examples in the high-numerical-aperture system demonstrate object reconstruction at high lateral resolution and retrieval of information in the depth direction of an object.