On the Propagation of Nonlinear Waves through a Medium Endowed with a Periodic Structure

It is a well-known fact that when waves exist in a periodic structure, if these waves’ wavelength is twice as long as the structure’s periodicity, they cannot stand steadily, and when they come upon the surface of the medium, they will reflect, being unable to penetrate into it. These phenomena are exhibited by all kinds of waves, not exclusively by a particular type wave, and therefore, they may be considered as basic characteristics of waves. Bloch’s theorem, which plays a crucial role in the theory of electrons in metals, and Bragg’s law of reflection in the X ray analysis of the crystalline structure relate to this fact. These phenomena can be considered as resulting from the interference between a wave going forward and a wave coming backward, the latter generated through reflection due to the periodic variation of medium’s property. Theoretically, these waves are usually treated with the Mathieu equation, which is also used to analyze the stability of the parametric oscillations. 1) However, in studies of this problem, it has usually been treated as a linear problem, and to the author’s knowledge, the problem of nonlinear effects upon these phenomena has to this time neither theoretically nor experimentally attracted the interest of researchers. However, it seems to be an interesting problem, not only in an academic sense but in a practical sense, to see, for instance, how light waves will propagate through a nonlinear medium with a periodic structure. Recently, Onogi and the present author 2) have studied a nonlinear Mathieutype equation and obtained interesting results regarding the nonlinear stability of the solution. The equation governing wave propagation through a medium endowed with a periodic structure resembles the nonlinear Mathieu equation. Hence, we can apply the method employed in the study of oscillation to the problem of wave propagation, and by the nonlinear stability criterion obtained there, we can hypothesize how waves