Semiclassical coupled wave theory for TM waves in one-dimensional photonic crystals

Semiclassical coupled wave theory is extended to waves with the electric field polarized parallel to the plane of incidence (TM waves) in one-dimensional periodic dielectric structures. Using this theory, the bandwidths and reflection/transmission characteristics of such structures, as functions of the incident wave frequency, are in good agreement with exact numerical simulations even for very high refractive index contrast.

Figure 1

Two-layered periodic dielectric structure (bi-layer photonic crystal).

Figure 2

Bloch phase and reflection vs frequency for TM mode at normal and 85° angle of incidence on the two-layered periodic structure with the parameters described in text; exact numerical results (solid line); first approximation of the semiclassical theory (thin line with points). The shaded region shows the omnidirectional band gap.

Figure 3

Harmonic periodic dielectric structure.

Figure 4

Bloch phase and reflection vs frequency for TM mode at normal and 85° angle of incidence on the harmonic periodic structure with a moderate $(n_A=0.5)$ refractive index modulation; the parameters are as described in text; exact numerical results (solid line); first approximation of the semiclassical theory (thin line with points).

Figure 5

Bloch phase and reflection vs frequency for TM mode at normal and 85° angle of incidence on the harmonic periodic structure with a deep $(n_A=1.5)$ refractive index modulation; the parameters are as described in text; exact numerical results (solid line); first approximation of the semiclassical theory (gray solid line); second approximation of the semiclassical theory (points).