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The design of large-scale mirror-rotation free-space optical cross-connect switches based on arrays of microelectromechanical torsion mirrors is considered. The layout of a compact switch is first presented. The parameters of the Gaussian beam that maximizes the port count for a given mirror turn angle is then identified, and the supporting optical system needed to create the desired beam is defined. Scaling laws for the optical path length needed for a given number of ports are then derived. Numerical simulations are used to verify the ideal configuration, and scaling laws are proposed for various departures from the ideal. It is shown that ideal operation can still be maintained when the mirrors are curved, and operating conditions that minimize the effect of mirror curvature are identified.