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The Dynamical-gap formation in Weyl semimetals modulated by intenseelliptically polarized light is addressed through the solution of thetime-dependent Schr\"odinger equation for the Weyl Hamiltonian via the Floquettheorem. The time-dependent wave functions and the quasi-energy spectrum of thetwo-dimensional Weyl Hamiltonian under normal incidence of ellipticallypolarized electromagnetic waves are obtained using a non-perturbative approach.In it, the Weyl equation is reduced to an ordinary second-order differentialMathieu equation. It is shown that the stability conditions of the Mathieufunctions are directly inherited by the wave function resulting in aquasiparticle spectrum consisting of bands and gaps determined by dynamicaldiffraction and resonance conditions between the electron and theelectromagnetic wave. Estimations of the electromagnetic field intensity andfrequency, as well as the magnitude of the generated gap are obtained for the$8-Pmmn$ phase of borophene. We provide with a simple method that enables topredict the formation of dynamical-gaps of unstable wave functions and theirmagnitudes. This method can readily be adapted to other Weyl semimetals.

Dynamical band gap tuning in Weyl semi-metals by intense elliptically polarized normal illumination and its application to $8-Pmmn$ borophene