  Recent advances in experimental control of colloidal systems have spurred a
revolution in the production of mesoscale thermodynamic devices. Functional
"textbook" engines, such as the Stirling and Carnot cycles, have been produced
in colloidal systems where they operate far from equilibrium. Simultaneously,
significant theoretical advances have been made in the design and analysis of
such devices. Here, we use methods from thermodynamic geometry to characterize
the optimal finite-time, nonequilibrium cyclic operation of the parametric
harmonic oscillator contact with a time-varying heat bath, with particular
focus on the Brownian Carnot cycle. We derive the optimally parametrized Carnot
cycle, along with two other new cycles and compare their dissipated energy,
efficiency, and steady-state power production against each other and a
previously tested experimental protocol for the Carnot cycle. We demonstrate a
20\% improvement in dissipated energy over previous experimentally tested
protocols and a $\sim$50\% improvement under other conditions for one of our
engines, while our final engine is more efficient and powerful than the others
we considered. Our results provide the means for experimentally realizing
optimal mesoscale heat engines.
