Properties of translation operator and the solution of the eigenvalue and boundary value problems of arbitrary space–time periodic metamaterials

There is a recent interest in understanding and exploiting the intriguing properties of space–time metamaterials. In the current manuscript, the time periodic circuit theory is exploited to introduce an appropriate translation operator that fully describes arbitrary space–time metamaterials. It is shown that the underlying mathematical machinery is identical to the one used in the analysis of linear time invariant periodic structures, where time and space eigen-decompositions are successively employed. We prove some useful properties the translation operator exhibits. The wave propagation inside the space time periodic metamaterial and the terminal characteristics can be rigorously determined via the expansion in the operators eigenvectors (space–time Bloch waves). Two examples are provided that demonstrate how to apply the framework. In the first, a space time modulated composite right left handed transmission line is studied and results are verified via time domain computations. Furthermore, we apply the theory to explain the non-reciprocal behaviour observed on a nonlinear transmission line manufactured in our lab. Bloch-waves are computed from the extracted circuit parameters. Results predicted using the developed machinery agree with both measurements and time domain analysis. Although the analysis was carried out for electric circuits, the approach is valid for different domains such as acoustic and elastic media.