Launching of gap solitons in nonuniform gratings.

Theoretical studies of the properties of nonlinear grating structures'3 have predicted the existence of gap solitons in these media. Recent experiments by Sankey et al. 4 confirm some of these predictions; in particular, they observe all-optical switching behavior' in silicon waveguide structures. The switching occurs between a low-transmissivity state, in which the structure essentially behaves as if it were linear, and a strongly nonlinear hightransmissivity state, in which a gap soliton is present. In these experiments use is made of the nonlinearity owing to the free carriers that are created by the light. Because the size of this nonlinearity is large (of the order of -10-14 m 2/W), Sankey et al. 4 were not limited in their experiments by the available optical power. However, because free carriers are involved, the decay of the nonlinearity is of the order of nanoseconds. Because this is of the same order as the pulse width, some effects predicted theoretically for an instantaneous Kerr medium cannot be confirmed by using this geometry. In other possible experiments use would be made of the electronic nonlinearity in semiconductors. 5 Though such nonlinearities have a femtosecond response time, they are also much smaller than those that are due to free carriers, typically by more than 3 orders of magnitude. If use is made of such nonlinearities, then the externally available power does become the overriding issue, and it is therefore crucial to design the structure so as to minimize input power requirements. The generation of gap solitons requires a powerful external source because their frequency content is close to the Bragg frequency of the grating. At low intensities, therefore, the radiation is strongly reflected. To launch a gap soliton, the external power must be sufficiently strong to weaken this reflection. The strong reflection in the linear limit can also be interpreted as resulting from a large impedance mismatch between the incoming medium and the periodic structure. Such problems are common in, for example, thin-film optics,6 and it is well known that the inclusion of a suitably chosen buffer layer can change the characteristics dramatically. Indeed, one of us has shown earlier that the inclusion of a buffer layer can lower the threshold for gap soliton formation. 7 However, the refractive index of such a layer must often differ substantially from that of the average refractive index of the grating, and layers with such different indices may not be easy to apply. Haus' has recently proposed using a second grating to improve the impedance matching, and thus to facilitate soliton launching. While his approach appears most applicable to solitons constructed from frequencies outside the photonic band gap, our interest is in launching the more distinctly nonlinear fields with center frequencies inside the gap. Impedance matching can also be obtained by making the grating nonuniform by applying, for example, a taper or a chirp. Here we describe a procedure to select an appropriate nonuniformity for the task at hand. To choose a nonuniformity, it is crucial to realize that one desires the largest possible energy density inside the structure when an external pulse is incident. We take the field in the leading edge of the pulse to vary as exp(-iwot). Here wc 0 is a complex frequency, with an imaginary part corresponding to the growth rate of the pulse. Certainly the total energy inside the grating is largest if no light is reflected; hence we must require