  Gravitational waves can act like gravitational lenses, affecting the observed
positions, brightnesses, and redshifts of distant objects. Exact expressions
for such effects are derived here in general relativity, allowing for
arbitrarily-moving sources and observers in the presence of plane-symmetric
gravitational waves. At least for freely falling sources and observers, it is
shown that the commonly-used predictions of linear perturbation theory can be
generically overshadowed by nonlinear effects; even for very weak gravitational
waves, higher-order perturbative corrections involve secularly-growing terms
which cannot necessarily be neglected when considering observations of
sufficiently distant sources. Even on more moderate scales where linear effects
remain at least marginally dominant, nonlinear corrections are qualitatively
different from their linear counterparts. There is a sense in which they can,
for example, mimic the existence of a third type of gravitational wave
polarization.
