Complex beams play important roles in wireless communications, radar, and satellites, and have attracted great interest in recent years. In light of this background, we present a fast and efficient approach to realize complex beams by using semidefinite relaxation (SDR) optimization and amplitude-phase digital coding metasurfaces. As the application examples of this approach, complex beam patterns with cosecant, flat-top, and double shapes are designed and verified using full-wave simulations and experimental measurements. The results show excellent main lobes and low-level side lobes and demonstrate the effectiveness of the approach. Compared with previous works, this approach can solve the complex beam-forming problem more rapidly and effectively. Therefore, the approach will be of great significance in the design of beam-forming systems in wireless applications.

复杂波束在无线通信、雷达和卫星中发挥着重要作用，近年来引起了人们的极大兴趣。鉴于这种背景，我们提出了一种快速有效的方法，通过使用半定弛豫（SDR）优化和幅度相位数字编码超表面来实现复杂光束。作为该方法的应用示例，使用全波模拟和实验测量设计并验证了具有余割、平顶和双形状的复杂光束图案。结果显示出出色的主瓣和低电平旁瓣，并证明了该方法的有效性。与之前的工作相比，该方法可以更快、更有效地解决复杂的波束形成问题。因此，该方法对于无线应用中波束形成系统的设计具有重要意义。