  The aim of this article is to advance the field of metamaterials by proposing
formulas for the design of high-contrast metamaterials with prescribed
subwavelength defect mode eigenfrequencies. This is achieved in two settings:
(i) design of non-hermitian static materials and (ii) design of instantly
changing non-hermitian time-dependent materials. The design of static materials
is achieved via characterizing equations for the defect mode eigenfrequencies
in the setting of a defected dimer material. These characterizing equations are
the basis for obtaining formulas for the material parameters of the defect
which admit given defect mode eigenfrequencies. Explicit formulas are provided
in the setting of one and two given defect mode eigenfrequencies in the setting
of a defected chain of dimers. In the time-dependent case, we first analyze the
influence of time-boundaries on the subwavelength solutions. We find that
subwavelength solutions are preserved if and only if the material parameters
satisfy a temporal Snell's law across the time boundary. The same result also
identifies the change of the time-frequencies uniquely. Combining this result
with those on the design of static materials, we obtain an explicit formula for
the material design of instantly changing defected dimer materials which admit
subwavelength modes with prescribed time-dependent defect mode eigenfrequency.
Finally, we use this formula to create materials which admit spatio-temporally
localized defect modes.
