Addition Theorem for Digital Coding Metamaterials

Coding representation of metamaterials builds up a bridge between the physical world and the digital world, making it possible to manipulate electromagnetic (EM) waves by digital coding sequences and reach field-programmable metamaterials. Here, the coding space is extended to complex domain and proposed complex digital codes to provide closer essence of EM-wave propagation. Based on the analytic geometry and complex variable functions, an addition theorem on complex coding is established, which reveals inherent connections among digital codes with different bits and enables all higher-bit digital codes to be represented by the 1-bit complex codes. According to the complex coding and addition theorem, multifunctional metamaterials can be directly designed and realized without considering mutual coupling. When two different coding patterns with different functions are added together via the addition theorem in complex form, the combined coding pattern will directly generate the two functions simultaneously without any perturbations. A series of realistic coding metasurfaces is presented to demonstrate the powerful and flexible performance of the complex coding and addition theorem for independent controls of EM waves to reach multiple functions. Good agreements between numerical simulations and experimental results prove the feasibility of the proposed concept and theorem in practical applications.