We present results from deep, wideband, high spatial and spectral resolution observations of the nearby luminous radio galaxy Cygnus A with the Jansky Very Large Array. The high surface brightness of this source enables detailed polarimetric imaging, providing images at $0.75arcsec$, spanning 2 - 18 GHz, and at 0.30$arcsec$ (6 - 18 GHz). The fractional polarization from 2000 independent lines of sight across the lobes decreases strongly with decreasing frequency, with the eastern lobe depolarizing at higher frequencies than the western lobe. The depolarization shows considerable structure, varying from monotonic to strongly oscillatory. The fractional polarization in general increases with increasing resolution at a given frequency, as expected. However, there are numerous lines of sight with more complicated behavior. We have fitted the $0.3arcsec$ images with a simple model incorporating random, unresolved fluctuations in the cluster magnetic field to determine the high resolution, high-frequency properties of the source and the cluster. From these derived properties, we generate predicted polarization images of the source at lower frequencies, convolved to 0.75$arcsec$. These predictions are remarkably consistent with the observed emission. The observations are consistent with the lower-frequency depolarization being due to unresolved fluctuations on scales $gtrsim$ 300 - 700 pc in the magnetic field and/or electron density superposed on a partially ordered field component. There is no indication in our data of the location of the depolarizing screen or the large-scale field, either, or both of which could be located throughout the cluster, or in a boundary region between the lobes and the cluster.
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