  Magnetic exchange couplings can be tuned by coupling to periodic light, where
the frequency and amplitude are typically varied: a process known as Floquet
engineering. The polarization of the light is also important, and in this
paper, we show how different polarizations, including several types of
unpolarized light, can tune the exchange couplings in distinct ways. Using
unpolarized light, for example, it is possible to tune the material without
breaking either time-reversal or any lattice symmetries. To illustrate these
effects generically, we consider single-band Hubbard models at half-filling on
the honeycomb, square and triangular lattices. We derive the effective
Heisenberg spin models to fourth order in perturbation theory for arbitrary
fixed polarizations, and several types of unpolarized light that preserve
time-reversal and lattice symmetries. Coupling these models to periodic light
tunes first, second and third neighbor exchange couplings, as well as ring
exchange terms on the square and triangular lattices. Circularly polarized
light induces chiral fields for the honeycomb and triangular lattices, which
favors non-coplanar magnetism and potential chiral spin liquids. We discuss how
to maximize the enhancement of the couplings without inducing heating.
