  Geometrical optics (GO) is often used to model wave propagation in weakly
inhomogeneous media and quantum-particle motion in the semiclassical limit.
However, GO predicts spurious singularities of the wavefield near reflection
points and, more generally, at caustics. We present a new formulation of GO,
called metaplectic geometrical optics (MGO), that is free from these
singularities and can be applied to any linear wave equation. MGO uses
sequenced metaplectic transforms of the wavefield, corresponding to symplectic
transformations of the ray phase space, such that caustics disappear in the new
variables, and GO is reinstated. The Airy problem and the quantum harmonic
oscillator are studied analytically using MGO for illustration. In both cases,
the MGO solutions are remarkably close to the exact solutions and remain finite
at cutoffs, unlike the usual GO solutions.
