1：Experimental Design and Method Selection:

The methodology involves the use of machine learning inverse design based on multilayer perceptron artificial neural networks (MANN) for the time-efficient and accurate reconstruction of a 3D vectorial holographic image. The principle involves designing a vectorial hologram with two digital functions of a phase hologram and a 2D vector field distribution to reconstruct a 3D vectorial holographic image.

2：Sample Selection and Data Sources:

The samples used include gold nanorods for fluorescence imaging to verify the three orthogonal components of a 3D vectorial field.

3：List of Experimental Equipment and Materials:

A spatial light modulator (SLM) with a split screen was used to prepare the MANN-derived 2D vector fields. High-resolution laser-printed phase patterns in a transparent photoresist were used for 3D direct laser writing of a vectorial hologram.

4：Experimental Procedures and Operational Workflow:

The process involves training the MANN to produce a 2D vector field through synthesizing the weighted amplitude and phase of the derived azimuthal and radial spatial components. The experimental verification includes two-photon fluorescence imaging of anisotropic single gold nanorods.

5：Data Analysis Methods:

The approach for analyzing experimental data includes statistical techniques and software tools for evaluating the electric field components of a 3D vectorial field.