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A three-dimensional theory of nonrelativistic, laminar, space-charge-limited, ellipse-shaped, charged-particle beam formation has been developed recently. Charged particles (electrons or ions) are accelerated across a diode by a static voltage differential and focused transversely by Pierce-type external electrodes placed along analytically specified surfaces. The theory is presented here along with design examples for a 6:1 elliptic electron beam diode and a 3:2 heavy ion beam diode. OMNITRAK simulations are performed to verify the theory and to explore beam tolerances with respect to perturbations of the diode geometry of the sort likely to be encountered in a realistic device: finite electrode extent, part misalignment, and allowances for thermal isolation.