A note on optical parametric backward-wave instabilities

The Fourier-Laplace integral representation for the field quantities involved in a backward-wave parametric instability is evaluated asymptotically by first performing the integration over ω and then performing a saddle-point asymptotic evaluation of the k integral on the appropriate sheet of ω(k), the dispersion relation for the parametric medium. The existence of time-growing solutions in an unbounded parametric medium is predicted, and the result shows spatial dependence related solely to the location of the saddle point in the k plane.