Virtual VNA 3.1: Non-Coherent-Detection-Based Non-Reciprocal Scattering Matrix Estimation Leveraging a Tunable Load Network

We refine the recently introduced "Virtual VNA 3.0" technique to remove the need for coherent detection. The resulting "Virtual VNA 3.1" technique can unambiguously estimate the full scattering matrix of a non-reciprocal, linear, passive, time-invariant device under test (DUT) with $N$ monomodal ports using an $N_\mathrm{A}$-channel coherent wavefront generator and an $N_\mathrm{A}$-channel non-coherent detector, where $N_\mathrm{A}<N$. Waves are injected and received only via a fixed set of $N_\mathrm{A}$ "accessible" DUT ports while the remaining $N_\mathrm{S}$ "not-directly-accessible" DUT ports are terminated by a specific tunable load network. To resolve all ambiguities, an additional modified setup is required in which waves are injected and received via a known $2N_\mathrm{A}$-port system connected to the DUT's accessible ports. We experimentally validate our method for $N_\mathrm{A}=N_\mathrm{S}=4$ considering a non-reciprocal eight-port circuit as DUT. By eliminating the need for coherent detection, our work reduces the hardware complexity which may facilitate applications to large-scale or higher-frequency systems. Additionally, our work provides fundamental insights into the minimal requirements to fully and unambiguously characterize a non-reciprocal DUT.