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The gravitational field of supermassive black holes is able to strongly bendlight rays emitted by nearby sources. When the deflection angle exceeds $\pi$,gravitational lensing can be analytically approximated by the so-called strongdeflection limit. In this paper we remove the conventional assumption ofsources very far from the black hole, considering the distance of the source asan additional parameter in the lensing problem to be treated exactly. We findexpressions for critical curves, caustics and all lensing observables valid forany position of the source up to the horizon. After analyzing the sphericallysymmetric case we focus on the Kerr black hole, for which we present ananalytical 3-dimensional description of the higher order caustic tubes.

Strong deflection limit of black hole gravitational lensing with arbitrary source distances