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A variational principle for the frequency of electromagnetic systems with radiation is derived from Maxwell's equations. Expressions for the quality factor are obtained in three illustrative examples: a leaky transmission line resonator, a grating coupler, and a curved slab waveguide. Extensive numerical results for the of the grating coupler are presented in normalized form. By optimization of the variational expressions with respect to the amplitudes of the forward and backward waves, coupled-mode equations are derived for a grating structure. The variational formulas agree with results obtained from a perturbational analysis.