Parametric representation of traveling waves in periodic media

As shown in our previous works, the linear oscillator equation with varying eigenfrequency can be integrated explicitly by introducing a new independent variable defined as the phase of non-harmonic oscillations. Such a parametrization considerably simplifies the theory of parametric resonance. By analogy, this analytical result describes exponential wave field damping in 1D photonic crystal band-gaps and can be used for optimization purposes. A similar approach can be applied to the transmission bands of a periodical medium. However, the problem proves to be more difficult compared with the stop-bands. To overcome the difficulty, one has to correctly define the phase of a complex wave function. We discuss the analytical relations following from the integrability conditions and particular solutions of the corresponding differential equations.