  In this work we use Floquet theory to theoretically study the influence of
monochromatic circularly and linearly polarized light on the Hofstadter
butterfly---induced by a uniform perpendicular magnetic field--for both the
kagome and triangular lattices. In the absence of the laser light, the
butterfly has fractal structure with inversion symmetry about magnetic flux
$\phi=1/4$, and reflection symmetry about $\phi=1/2$. As the system is exposed
to an external laser, we find circularly polarized light deforms the butterfly
by breaking the mirror symmetry at flux $\phi=1/2$. By contrast, linearly
polarized light deforms the original butterfly while preserving the mirror
symmetry at flux $\phi=1/2$. We find the inversion symmetry is always preserved
for both linear and circular polarized light. For linearly polarized light, the
Hofstadter butterfly depends on the polarization direction. Further, we study
the effect of the laser on the Chern number of lowest band in the off-resonance
regime (laser frequency is larger than the bandwidth). For circularly polarized
light, we find that low laser intensity will not change the Chern number, but
beyond a critical intensity the Chern number will change. For linearly
polarized light, the Chern number depends on the polarization direction. Our
work highlights the generic features expected for the periodically driven
Hofstadter problem on different lattices.
