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    Area of Science:

    • Quantum mechanics
    • Photonics
    • Wave function reconstruction

    Background:

    • The wave function is crucial for predicting photon behavior.
    • Weak measurement allows wave function reconstruction, but current methods lack completeness.
    • Existing relationships between wave function, weak value, and probes are limited.

    Purpose of the Study:

    • To propose a novel wave function reconstruction method.
    • To establish complete relationships between wave function, complex weak values, and complex shift probes.
    • To enhance applications in quantum imaging and diagnostics.

    Main Methods:

    • Utilizing complex weak values derived from pre- and post-selection in weak measurements.
    • Employing complex shift probes based on Fresnel coefficients in total internal reflection.
    • Developing a wave function reconstruction technique using position and momentum shift probes.

    Main Results:

    • Four novel relationships between wave function, weak values, and probes were established.
    • Successful experimental reconstruction of phase distribution using a real weak value and momentum shift probe.
    • Experimental results demonstrated high contrast and agreement with theoretical predictions.

    Conclusions:

    • The proposed scheme offers a convenient method for wave function reconstruction.
    • The study completes the understanding of relationships between wave function, weak values, and probes.
    • This work provides a valuable reference for applications in image processing, quantitative phase imaging, and label-free biological imaging.