Application of a variational principle to systems with radiation loss

A variational principle for the frequency of electromagnetic systems with radiation is derived from Maxwell's equations. Expressions for the quality factor
Q
are obtained in three illustrative examples: a leaky transmission line resonator, a grating coupler, and a curved slab waveguide. Extensive numerical results for the
Q
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.