Radars are an ideal complement to cameras: both are inexpensive, solid-state sensors, with cameras offering fine angular resolution, while radars provide metric depth and robustness under adverse weather. However, radar data is more difficult to interpret than camera images and varies significantly between sensors, necessitating increased reliance on simulation for prototyping sensors and processing pipelines. Recent work treating radar reconstruction as a novel view synthesis problem has shown great promise in reconstructing radar-relevant geometry and simulating low-level radar data. However, such methods are constrained by the low spatial resolution of the underlying radar. To address this, we propose a unified differentiable renderer, RadarSim, which leverages the high angular resolution of RGB cameras to generate Doppler radar range images from a camera-initialized neural field. Using a novel data set of calibrated radar camera recordings from a custom hand-held rig, we demonstrate that RadarSim produces sharper geometry and Doppler range frames than radar-only reconstructions.