Phase-retrieval algorithm based on Kramers–Kronig relations in coherent diffraction imaging

Coherent diffraction imaging (CDI) is a high-resolution technique that does not require x-ray lenses. With advances in scientific technology, such as synchrotron radiation, x-ray free-electron lasers, and coherent electron sources, CDI has been applied to diverse fields, such as biology, medicine, and semiconductors, as a high-resolution, nondestructive measure. With the rapid increase in demand for these applications, enhancing the efficiency of processing high-volume data has become a significant challenge for promotion. In this study, we proposed an algorithm that combines Kramers–Kronig (KK) relations with oversampling smoothness (OSS). The results were evaluated by introducing an error coefficient. We found that the error of the KK-OSS algorithm is always reduced by approximately 50% compared with the error reduction (ER) algorithm, hybrid input–output (HIO) algorithm, and OSS in real space. In the diffraction space, the error in the KK-OSS can be decreased to 0.11. With 100 iterations, KK-OSS spent 218.3 s on reconstructing most of the sample information, while ER was 258.1 s, HIO algorithm took 296.7 s and the reconstruction was still a random value. In Fraunhofer diffraction, it cost KK-OSS 58.8 s to reconstruct, while OSS took 61.9 s. Therefore, this method can reduce the reconstruction error, shorten the reconstruction time, and improve the efficiency compared with the ER, HIO, and OSS algorithms using a random phase as the initial value.