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Synthetic aperture sonar (SAS) requires precise positional and environmental information to produce well-focused output during the image reconstruction step. However, errors in these measurements are commonly present resulting in defocused imagery. To overcome these issues, an \emph{autofocus} algorithm is employed as a post-processing step after image reconstruction for the purpose of improving image quality using the image content itself. These algorithms are usually iterative and metric-based in that they seek to optimize an image sharpness metric. In this letter, we demonstrate the potential of machine learning, specifically deep learning, to address the autofocus problem. We formulate the problem as a self-supervised, phase error estimation task using a deep network we call Deep Autofocus. Our formulation has the advantages of being non-iterative (and thus fast) and not requiring ground truth focused-defocused images pairs as often required by other deblurring deep learning methods. We compare our technique against a set of common sharpness metrics optimized using gradient descent over a real-world dataset. Our results demonstrate Deep Autofocus can produce imagery that is perceptually as good as benchmark iterative techniques but at a substantially lower computational cost. We conclude that our proposed Deep Autofocus can provide a more favorable cost-quality trade-off than state-of-the-art alternatives with significant potential of future research.