Investigating non-reciprocity in time-periodic media using a perturbative approach

Lorentz famous theorem leads to clear reciprocity conditions for linear, time-invariant media based on their constitutive parameters. By contrast, reciprocity conditions for linear time-varying media are not fully explored. In this paper, we investigate whether, and how a structure containing a time-periodic medium can be truly identified as reciprocal or not. To that end, a necessary and sufficient condition is derived which requires both the constitutive parameters and the electromagnetic fields inside the dynamic structure. As solving for the fields for such problems is challenging, a perturbative approach is proposed which expresses the aforementioned non-reciprocity condition in terms of the electromagnetic fields and the Green's functions of the unperturbed static problem and is particularly applicable for the case of structures with weak time modulation. Reciprocity of two famous canonical time-varying structures are then studied using the proposed approach and their reciprocity/non-reciprocity is investigated. In the case of one-dimensional propagation in a static medium with two point-wise modulations, our proposed theory clearly explains the often observed maximization of non-reciprocity when the modulation phase difference between the two points is 90 degrees. In order to validate the perturbative approach, analytical and Finite-Difference Time-Domain (FDTD) methods are employed. Then, solutions are compared and considerable agreement between them is observed.