In this paper, we consider random phase fluctuations imposed during wave propagation through a turbulent plasma (e.g. ionosphere) as a source of additional noise in interferometric visibilities. We derive expressions for visibility variance for the wide field of view case (FOV$\sim10$ deg) by computing the statistics of Fresnel diffraction from a stochastic plasma, and provide an intuitive understanding. For typical ionospheric conditions (diffractive scale $\sim 5-20$ km at $150$ MHz), we show that the resulting ionospheric `scintillation noise' can be a dominant source of uncertainty at low frequencies ($\nu \lesssim 200$ MHz). Consequently, low frequency widefield radio interferometers must take this source of uncertainty into account in their sensitivity analysis. We also discuss the spatial, temporal, and spectral coherence properties of scintillation noise that determine its magnitude in deep integrations, and influence prospects for its mitigation via calibration or filtering.
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