P.T.D and Rotational Symmetries in Non-Reflecting Structures

$\mathcal{P}\cdot\mathcal{T}\cdot\mathcal{D}$ and rotational symmetries of electromagnetic and photonic systems have recently been shown, separately, to characterize different non-reflecting structures. In this work, we provide a general framework for both types of symmetries, casting the rotational symmetry in terms of a new $\mathcal{R}\cdot\mathcal{T}\cdot\mathcal{D}$ - type symmetry. Out of this analysis, a third option emerges; namely, a combined $\mathcal{P}\cdot\mathcal{T}\cdot\mathcal{D}/\mathcal{R}\cdot\mathcal{T}\cdot\mathcal{D}$ symmetry condition. With this option, the door opens for synthesis of many new non-reflecting media, as shown in the examples below.