Floquet Engineering of Topological Phases and Magneto-Optical Response in a Driven $d$-wave Altermagnet

We study how Floquet driving with linearly polarized light controls the topology and magneto-optical response of a two-dimensional (2D) $d$-wave altermagnet. In the absence of linearly polarized optical field and under spin conservation, we find that the system hosts a spin-Chern (a quantum-spin-Hall analog) phase with Chern numbers of opposite sign in the two spin sectors. The irradiated optical field breaks the $C_{4z}\mathcal{T}$ crystalline antiunitary symmetry between the spin sectors. Symmetry breaking originates from polarization-dependent Peierls phases, which renormalize hopping anisotropically along the two axes. The resulting spin-selective gap closures produce intermediate Chern-insulating phases with $C=\pm1$. The drive amplitude $A_0$ determines the inversion thresholds, while rotating the polarization by $\pi/2$ swaps the spin sectors and reverses the Chern number. Using the Kubo formalism, we compute the frequency-dependent longitudinal and Hall conductivities and derive the corresponding Faraday and Kerr rotations for a free-standing conducting sheet. The longitudinal response tracks the Floquet-renormalized interband thresholds, whereas the optical Hall response, together with the sign of the magneto-optical rotations, distinguishes the two opposite Berry-curvature chiralities. Sizable Kerr angles occur only within narrow resonant windows and should be interpreted together with the reflected intensity and Kerr ellipticity. These results identify linearly polarized light as a symmetry-selective handle for spin-resolved band inversion, Chern-number switching, and contact-free optical detection in $d$-wave altermagnets.