  The development of active metasurface systems, such as lasing metasurfaces,
requires the optimization of multiple modes at the absorption and lasing
wavelength bands, including their quality factor, mode profile and angular
dispersion. Often, these requirements are contradictory and impossible to
obtain with conventional design techniques. Importantly, the properties of the
eigenmodes of a metasurface are directly linked to their symmetry, which offers
an opportunity to explore mode symmetry as an objective in optimization
routines for active metasurface design. Here, we propose and numerically
demonstrate a novel multi-objective optimization technique based on symmetry
projection operators to quantify the symmetry of the metasurface eigenmodes. We
present, as an example, the optimization of a lasing metasurface based on
up-converting nano-particles. Our technique allows us to optimize the
absorption mode dispersion, as well as the directionality of the lasing
emission and therefore offers advantages for novel lasing systems with high
directionality and low lasing threshold.
