Analytic and passivity properties of reflection and transmission coefficients of thin-film multilayered stacks are investigated. Using a rigorous formalism based on the inverse Helmholtz operator, properties associated with the causality principle and passivity are established when both the temporal frequency and spatial wave vector are continued in the complex plane. This result extends the range of situations where the Kramers-Kronig relations can be used to deduce the phase from the intensity. In particular, it is rigorously shown that the Kramers-Kronig relations for reflection and transmission coefficients remain valid for all fixed angles of incidence. Possibilities for exploiting the new relationships are discussed and numerically tested.