Light-modulated Josephson effect in Kekul\'e patterned graphene

We theoretically study the Josephson effect in a superconductor/normal metal/superconductor junction based on Kekul'e patterned graphene. For the Kekul'e-O patterned junctions, a Fermi momentum-splitting Andreev reflection at the interface can be induced by the off-resonant circularly polarized light applied in the normal region, which results in the possible $\pi$-state. In contrast, for the Kekul'e-Y patterned junctions, the Fermi momentum-splitting Andreev reflection is strongly suppressed due to the valley-momentum locking and the junction always exhibits the 0-state. The dependence of the critical current on the junction length and the illumination parameter of the light field is also presented in detail.