  The upper bound on antenna gain is expressed as a sum of lossy characteristic
modes, specifically, as a sum of characteristic far fields squared. The
procedure combines the favorable properties of Harrington's classical approach
to maximum directivity and current-density-based approaches. The upper bound is
valid for any antenna or array designed in a given design region for which
optimal performance is determined. The decomposition into modes makes it
possible to study the degrees of freedom of an obstacle, classify its radiation
into normal or super-directive currents, and determine their compatibility with
a given excitation. The bound considers an arbitrary shape of the design region
and specific material distribution. The cost in Q-factor and radiation
efficiency is studied. The extra constraint of a self-resonance current is
imposed for an electrically small antenna. The examples verify the developed
theory, demonstrate the procedure's utility, and provide helpful insight to
antenna designers. The feasibility of the optimal gain is studied in detail on
an example of end-fire arrays using the aforementioned decomposition of optimal
current density into lossy characteristic modes.
