  We exhibit the form of the ``radiation field,'' describing the large-scale,
long-time behavior of solutions to the wave equation on a manifold with no
trapped rays, as a Fourier integral operator. We work in two different
geometric settings: scattering manifolds (a class which includes asymptotically
Euclidean spaces) and asymptotically hyperbolic manifolds.
  The canonical relation of the radiation field operator is a map from the
cotangent bundle of the manifold to a cotangent bundle over the boundary at
infinity; it is associated to a sojourn time, or Busemann function, for
geodesic rays. In non-degenerate cases, the symbol of the operator can be
described explicitly in terms of the geometry of long-time geodesic flow.
  As a consequence of the above result, we obtain a description of the
(distributional) high-frequency asymptotics of the scattering-theoretic Poisson
operator, better known as the Eisenstein function in the asymptotically
hyperbolic case.
