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The optic nerve head (ONH) typically experiences complex neural- and connective-tissue structural changes with the development and progression of glaucoma, and monitoring these changes could be critical for improved diagnosis and prognosis in the glaucoma clinic. The gold-standard technique to assess structural changes of the ONH clinically is optical coherence tomography (OCT). However, OCT is limited to the measurement of a few hand-engineered parameters, such as the thickness of the retinal nerve fiber layer (RNFL), and has not yet been qualified as a stand-alone device for glaucoma diagnosis and prognosis applications. We argue this is because the vast amount of information available in a 3D OCT scan of the ONH has not been fully exploited. In this study we propose a deep learning approach that can: \textbf{(1)} fully exploit information from an OCT scan of the ONH; \textbf{(2)} describe the structural phenotype of the glaucomatous ONH; and that can \textbf{(3)} be used as a robust glaucoma diagnosis tool. Specifically, the structural features identified by our algorithm were found to be related to clinical observations of glaucoma. The diagnostic accuracy from these structural features was $92.0 \pm 2.3 \%$ with a sensitivity of $90.0 \pm 2.4 \% $ (at $95 \%$ specificity). By changing their magnitudes in steps, we were able to reveal how the morphology of the ONH changes as one transitions from a `non-glaucoma' to a `glaucoma' condition. We believe our work may have strong clinical implication for our understanding of glaucoma pathogenesis, and could be improved in the future to also predict future loss of vision.