Table 1 Key mathematical formulations.

Floquet-Bloch Field

\(E(x) = e^{i k x} \sum _{n} \hat{E}_n e^{i n \frac{2\pi }{d} x}\)

RIS-Modulated Field

\(E_{\text {mod}}(x) = e^{i \theta (x)} \sum _{n} \hat{E}_n e^{i \left( k + n \frac{2\pi }{d} \right) x}\)

Beam Steering Law

\(\sin \theta _r = \sin \theta _i + \frac{\beta \lambda }{2\pi }\)

Time-Modulated RIS

\(E_{\text {mod}}(x,t) = e^{i \beta x} \sum _{p=-\infty }^{\infty } J_p(\Delta \theta ) e^{i p \Omega t} \sum _{n=-\infty }^{\infty } \hat{E}_n e^{i \left( k + n \frac{2\pi }{d} \right) x}\)

Floquet MIMO Channel

\(\textbf{H} = \sum _{n} \textbf{H}_n e^{i n \frac{2\pi }{d} x}\)

Channel Loss Model

\(\textbf{H}_{\text {loss}} = \sum _{n} \textbf{H}_n e^{i n \frac{2\pi }{d} x} \cdot L(\theta _n, f)\)