Analog physics has successfully tackled the problems of gauge theories, event horizons, Big Bang and Universe expansion, and many others. Here, we suggest a photonic model system for a “time machine” based on the paraxial beam approximation. We demonstrate how the closed timelike curves and the well-known grandfather paradox can be studied experimentally in this system. We show how Novikov's self-consistency principle is realized in quantum mechanics owing to Heisenberg's uncertainty principle.