  Floquet engineering or coherent time periodic driving of quantum systems has
been successfully used to synthesize Hamiltonians with novel properties. In
ultracold atomic systems, this has led to experimental realizations of
artificial gauge fields, topological band structures, and observation of
dynamical localization, to name just a few. Here we present a Floquet-based
framework to stroboscopically engineer Hamiltonians with spatial features and
periodicity below the diffraction limit of light used to create them by
time-averaging over various configurations of a 1D optical Kronig-Penney (KP)
lattice. The KP potential is a lattice of narrow subwavelength barriers spaced
by half the optical wavelength ($\lambda/2$) and arises from the non-linear
optical response of the atomic dark state. Stroboscopic control over the
strength and position of this lattice requires time-dependent adiabatic
manipulation of the dark state spin composition. We investigate adiabaticity
requirements and shape our time-dependent light fields to respect the
requirements. We apply this framework to show that a $\lambda/4$-spaced lattice
can be synthesized using realistic experimental parameters as an example,
discuss mechanisms that limit lifetimes in these lattices, explore candidate
systems and their limitations, and treat adiabatic loading into the ground band
of these lattices.
