| In theory, we show that oversampling the diffraction pattern of a finite specimen corresponds to generating a no-density region surrounding the electron density of the specimen. This no-density region can then be used to retrieve the phase information. By comparing the number of unknown-variables in real space with the number of equations in reciprocal space, we suggest that, given a diffraction pattern sampled at the Nyquist frequency, the phase problem is underdetermined by a factor of 2 for 1D, 2D, and 3D objects. By employing an iterative algorithm and an initial random phase set, we retrieve the phase from noise-free and noisy diffraction patterns of 2D and 3D complex-valued objects by using positivity constraints on the imaginary part of the objects and loose supports, with the over-sampling factor much less than 4 for 2D and 8 for 3D objects. We then successfully extend the oversampling technique to the phase retrieval of small specimens which are (a) perfect or imperfect crystals, or (b) have a repeated motif without orientational or translational regularity, or (c) are an unrepeated motif, such as an amorphous fragment, a single molecule, or a single biological cell. The oversampling technique thus greatly extends the specimen range of X-ray crystallography, but it imposes a high radiation dose to the specimen compared with Bragg-point-only methods. With the possible appearance in the future of X-ray free electron lasers (>10 12 photons and <200 femtoseconds per pulse), one may be able to circumvent the radiation damage problem in additional cases by recording diffraction patterns with single ultrashort X-ray pulses.; In experiment, we demonstrate for the first time that a soft X-ray diffraction pattern from a micron-size non-crystalline specimen, consisting of an array of gold dots, can be recorded and inverted to form a high-resolution image. The phase problem is solved by oversampling the diffraction pattern. Our technique requires no high- resolution X-ray optical elements or detectors. We believe that considerably higher resolution can be achieved. This technique can thus open a door for high-resolution three-dimensional structure determination of non-crystalline specimens in both material science and structural biology. |
