  We give a prescription for N=1 supersymmetrization of any (four-dimensional)
nonlinear electrodynamics theory with a Lagrangian density satisfying a
convexity condition that we relate to semi-classical unitarity. We apply it to
the one-parameter ModMax extension of Maxwell electrodynamics that preserves
both electromagnetic duality and conformal invariance, and its Born-Infeld-like
generalization, proving that duality invariance is preserved. We also establish
superconformal invariance of the superModMax theory by showing that its
coupling to supergravity is super-Weyl invariant. The higher-derivative
photino-field interactions that appear in any supersymmetric nonlinear
electrodynamics theory are removed by an invertible nonlinear superfield
redefinition.
