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We demonstrate the existence of generalized Aubry-Andr\'e self-duality in aclass of non-Hermitian quasi-periodic lattices with complex potentials. Fromthe self-duality relations, the analytical expression of mobility edges isderived. Compared to Hermitian systems, mobility edges in non-Hermitian onesnot only separate localized from extended states, but also indicate thecoexistence of complex and real eigenenergies, making it possible a topologicalcharacterization of mobility edges. An experimental scheme, based on opticalpulse propagation in synthetic photonic mesh lattices, is suggested toimplement a non-Hermitian quasi-crystal displaying mobility edges.

Generalized Aubry-Andr\'e self-duality and Mobility edges in non-Hermitian quasi-periodic lattices