# Continuous spatio-temporal synthesis of electromagnetic fields by projected space-time Fourier transform

## TL;DR

A new method called PST-FT allows rapid and efficient synthesis of complex four-dimensional electromagnetic fields over time.

## Contribution

The PST-FT method enables analytical control of high-dimensional EM fields without iterative algorithms or matrix inversion.

## Key findings

- The PST-FT method successfully synthesizes three complex time-varying microwave fields.
- The approach allows rapid synthesis of continuous 4D EM fields.
- Corollaries and limitations were derived by integrating the theory with signal processing.

## Abstract

The manipulation of electromagnetic (EM) waves is important in diverse fields such as microwave sensing and wireless communication. Spatio-temporal synthesis, in particular, has attracted growing interest in recent years. Several related approaches have been explored, but their huge computational burden confines them to low-dimensional EM field synthesis. The synthesis of four-dimensional (4D) EM fields remains a challenging problem. Here, we develop the projected space-time Fourier transform (PST-FT) method to analytically control the continuous evolution of the high-dimensional EM field with time. Without iterative algorithms or pseudo-inverse matrix, it allows rapid synthesis of continuous 4D EM fields. For verification, three complicated time-varying microwave fields are synthesized, including the 4D electric field radiated by other antennas, the multi-target continuously scanning field, and the time-varying shaped uniform field. Furthermore, several corollaries and fundamental limitations are derived by integrating the developed theory and signal processing theory. Our results demonstrate the proposed theory, with potential application in reproducing EM environments, wireless communication and holographic imaging.

Yunlong Du and colleagues present a general and rapid method to analytically control the continuous evolution of the electromagnetic field with time. This approach enables efficient reproduction of the four-dimensional electromagnetic environment.