Nonlinear optical effects in periodic media

Summary form only given. The key property of periodic media, those in which the refractive index is a periodic function of position, is that they exhibit a Bragg resonance. At the associated Bragg frequency, and at frequencies close to it, light cannot propagate through such media, and is strongly reflected instead. It is perhaps less well known that, close to the Bragg frequency, the group velocity of the light is strongly reduced due to multiple reflections off the grating rulings. Since in a fiber grating the group velocity varies between 0 and the speed of light in untreated fiber in a fraction of a nanometer, the dispersion is also very large. Thus, close to the Bragg frequency, pulses propagate slowly, but this is accompanied by severe pulse broadening. The slow propagation, combined with strong dispersion, leads to many interesting effects at high intensities, where the Kerr nonlinearity of glass is important. These include slow soliton propagation, pulse compression, short pulse generation, and all-optical switching, all of which have been observed in the laboratory, using fiber or semiconductor geometries. Other developments include the study of periodic media with a second order nonlinearity. This is driven partly because the grating affects the phase matching for second harmonic generation, and, more recently, because a second order nonlinearity can lead to a nonlinear phase shift, somewhat similar to that of a Kerr nonlinearity, but at lower intensities. I will review these and other theoretical and experimental developments, aiming to provide some insight into the fundamentals of this rapidly developing field.