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Optically thin dielectric slabs, in which a fully etched-through two-dimensional patterning is applied, are used to form high-Q optical cavities with modal volumes approaching the theoretical limit of a cubic half-wavelength. Resonant cavities are formed from local defect regions within the photonic lattice. Simple group theoretical techniques are developed to design cavities which support resonant modes with a particular polarization and radiation pattern. Numerical simulations using thedoi:10.1088/1464-4258/3/6/367 fatcat:akstn3ia45cabimmulqc5vzcmq