Abstract

Optical precursors determine how a wave that is initially well-defined builds up when it propagates in a dispersive medium. A rigorous uniform description can be accomplished by means of a modern asymptotic theory.¹ In a first analysis, that is, for time-domain local precursors, these waves are described in the Fourier-integral representation in terms of one Bessel function (Sommerfeld precursor) and the Airy integral (Brillouin precursor).² In this work we determine, by using an operatorial equivalence based on the dispersion relation and under the same local restriction, the evolution equations satisfied by the precursors. Because of their asymptotic nature, precursors appear naturally as self-similar solutions of these equations. The above operatorial equivalence further suggests a simple way of extending the problem to take into account a perturbative Kerr nonlinearity,ßI, where ^I is the intensity of the field and ß is a perturbation parameter. Although such effect is negligible in practice, it is instructive to see that the equations still remain tractable. In fact, similarity properties still exist and permit us to perform a simple perturbation scheme. Our results show that nonlinearity increases (decreases) the Sommerfeld precursor spreading for ß 0 (ß < 0) and yields a self-phase modulation of the Brillouin precursor.

© 1990 Optical Society of America