  We study via Carleman estimates the sharpest possible exponential decay for
{\it waveguide} solutions to the Laplace equation
$$(\partial^2_t+\triangle)u=Vu+W\cdot(\partial_t,\nabla)u,$$ and find a
necessary quantitative condition on the exponential decay in the
spatial-variable of nonzero waveguides solutions which depends on the size of
$V$ and $W$ at infinity.
