Wave function reconstruction based on complex weak values

The wave function is important for understanding and predicting the behavior of photons. Recent studies have shown that wave function can be reconstructed through weak measurement. But currently, the relationships of wave function, weak value, and probe are single and not complete. Here, we propose a wave function reconstruction method based on complex weak values and complex shift probes. Four relationships of them are built and the wave function can be reconstructed by position and momentum shift probes. The complex weak values can be obtained by adjusting the pre- and post-selection in weak measurement, while the complex shift probes originate from the complex nature of the Fresnel coefficient in total internal reflection. To demonstrate the feasibility of the proposed scheme, an experiment of phase distribution reconstruction is achieved by a real weak value and a momentum shift probe; the experimental results agree well with the theory and show a high contrast. Therefore, the wave function can be conveniently reconstructed by our scheme and the relationships between the wave function, weak values, and probes become more complete. This work provides a valuable reference for the application of weak measurement in image processing, quantitative phase imaging, and label-free biological imaging.