The standard way to understand gravitational lensing is via geometric optics. However, one can instead adopt a more field-centered point of view and employ Green functions for Maxwell's equations (or the linearized Einstein equation) in an appropriate spacetime. Combining these two perspectives leads to new insights into the foundations of geometric optics and its generalizations. It becomes clear how the usual geometric-optics ansatz (and its higher-order generalizations) arises in different ways in different physical scenarios, and why tails never appear in high-frequency approximations (despite being present). It also becomes possible to directly relate field properties with source properties, and to identify some quantities as being invariant under large classes of geometrical transformations.