Semiclassical Coupled Wave Theory for TM-waves in 1-D Photonic Crystals

A one dimensional periodic structure with high refractive index contrast can serve as a photonic crystal [1–5]. Among the common theoretical methods for wave motion in such structures, the coupled wave approach offers superior physical insight and gives simple analytical results in limiting cases. Unfortunately, the conventional coupled wave theory of Kogelnik [6–8] fails in the case of high refractive index contrast, which is required for a 1D photonic crystal. In a previous article [9] we extended the first approximation of the semiclassical coupled wave theory for normal propagation [10, 11] to the case of oblique incidence of TE waves. Using the Bogolyubov averaging method [12], we developed a second approximation for the same case. Our method allows for both variable amplitudes and variable (geometric-optics) phases in the counterpropagating waves. While it is analytically almost as simple as conventional coupled wave theory, our method is essentially exact for any achievable ratio (e.g. 1.5 : 4.6) of the indices of refraction of the layers comprising buildable devices. In this paper we extend our semiclassical coupled wave theory to the case of oblique propagation of TM electromagnetic waves. Both the first and the second approximations are extracted. Surprisingly, the first approximation works even better for the TM case than it did for TE waves. The second approximation gives practically exact results even for structures with a very deep modulation of the refractive index. Using our analytic expressions for the band edges, one can easily optimize the positions and widths of the forbidden zones, in order to fine tune photonic devices. In the following section, the semiclassical coupled-wave method for TM waves is developed. We obtain simple analytical expressions for the Bloch phase, which is a key parameter for determination of band structure, and for the reflection/transmission amplitudes. As an application, in Sec. III we find the optimal omnidirectional forbidden band of a bilayer periodic dielectric structure. In Sec. IV we show how to apply the theory to a periodic structure with a continuous harmonic profile of the refractive index. The conclusions are in Sec. V.