Light-modulated Josephson effect in Kekulé patterned graphene

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