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We present an exhaustive numerical investigation of the optical caustics ingravitational lensing by a spinning black hole for an observer at infinity.Besides the primary caustic, we examine higher order caustics, formed byphotons performing one or several loops around the black hole. Ourinvestigation covers the whole parameter space, including the black hole spin,its inclination with respect to the line of sight, the source distance, and thecaustic order. By comparing our results with the available analyticalapproximations, we find perfect agreement in their respective domains ofvalidity. We then prove that all caustics maintain their shape (a tube withastroidal cross-section) in the entire parameter space without suffering anytransitions to different caustic shapes. For nearly extremal spin, however,higher order caustics grow so large that their cross-sections at fixed radiiwind several times around the black hole. As a consequence, for each causticorder, the number of images ranges from 2 to 2(n+1), where n is the number ofloops spanned by the caustic. As for the critical curves, we note that for highvalues of the spin they develop a small dip on the side corresponding toprograde orbits.

Optical caustics of Kerr spacetime: the full structure