Figure 1

(Color online) Frequency dispersion of the real part of the Bloch wave number in a bilayer PC (solid line) and of homogenized wave number $k_0\left(d_1+d_2\right)\sqrt{⟨\epsilon E{E}^{\ast }⟩/⟨E{E}^{\ast }⟩}$ (dashed line). Layers with permittivities ${\epsilon }_1=1$ and ${\epsilon }_2=3$ have equal thicknesses $d_1=d_2$.Reuse & Permissions

Figure 2

The results of the exact solution of the 1D Maxwell equation: the electric field ${\left|E\right|}^2$ distribution in a PC slab at frequencies of the nearest to BG edges Fabri-Perot resonances: the solid and the dashed lines present the case below $\left[k_0\left(d_1+d_2\right)=1.55\right]$ and above $\left[k_0\left(d_1+d_2\right)=2.77\right]$ the first BG, respectively. Darker and lighter areas depict layers with higher, ${\epsilon }_2=3$, and lower, ${\epsilon }_1=1$, permittivities, $d_1=d_2$. It is assumed that the slab is in vacuum, ${\epsilon }_{ext}=1$.Reuse & Permissions

Figure 3

The frequency dependence of the fraction of the electric field energy in the high-permittivity layer for the low-contrast PC with $d_1/d_2=1.108$, ${\epsilon }_1=2$ and ${\epsilon }_2=3$. (These values of thicknesses and permittivities correspond to the parameters $K_0=1$ and $\alpha =0.05$, which are defined below). BG regions are shaded. The horizontal line corresponds to the uniform distribution of the field.Reuse & Permissions

Figure 4

(Color online) Frequency dispersion of the real part of the Bloch wave number in a bilayer PC (solid line) and of homogenized wave number $k_0\left(d_1+d_2\right)\sqrt{⟨\epsilon E{E}^{\ast }⟩/⟨E{E}^{\ast }⟩}$ (dotted line). Layers with permittivity values ${\epsilon }_1=1$ and ${\epsilon }_2=100$ have equal thicknesses, $d_1=d_2$.Reuse & Permissions

Figure 5

(Color online) Evolution of the band structure with the change in the parameter $\alpha$. Dotted $\left[k_0/K_0=n_1/\left(1-\alpha \right)\right]$ and dashed $\left[k_0/K_0=n_2/\left(1+\alpha \right)\right]$ curves correspond to the Fabri-Perot resonances $k_0{d}_1\sqrt{{\epsilon }_1}=\pi n_1$ and $k_0{d}_2\sqrt{{\epsilon }_2}=\pi n_2$, respectively. BG regions are shaded. The permittivity values of the layers are ${\epsilon }_1=1$ and ${\epsilon }_2=100$. The width of the second BG becomes zero at $k_0/K_0=1$.Reuse & Permissions

Figure 6

(Color online) The frequency dependence of the energy fraction in the high-permittivity layer (solid line) for different structures of the primitive cell at a high contrast PC (${\epsilon }_1=1$ and ${\epsilon }_2=100$). (a) $\alpha =0.15$ and (b) $\alpha =-0.15$. BG regions are shaded. The vertical lines show frequencies of the Fabry-Perot resonances that are shown by dashed and dotted lines in Fig. 5. The direct Borrmann effect occurs when of the energy fraction decreases with the frequency inside a BG. The inversion of this dependency corresponds to the inverse Borrmann effect. The latter phenomenon is observed when a pass band containing the Fabry-Perot resonance in the lower-permittivity layer (dotted vertical lines) is below the BG and a pass band containing higher-permittivity layer Fabry-Perot resonance (dashed vertical lines) is above the BG.Reuse & Permissions

Figure 7

(Color online) The transmittance (dashed line) and the normalized Faraday rotation (solid line) of a slab of a MPC for frequencies (a) below and (b) above the zero-width BG. BG regions are shaded.Reuse & Permissions

Figure 8

(Color online) Thick line is a fragment of the same dispersion curve as in Fig. 1. The straight lines are dispersion curves of nonperturbed medium at different choices of ${\epsilon }_0$. Line 1 corresponds to the homogenization wave vector $k_B=k_0\sqrt{⟨\epsilon ⟩}$, line 2 to the exact theory dependence $k_B=k_0⟨\sqrt{\epsilon }⟩$.Reuse & Permissions

Figure 9

The frequency dependence of the energy fraction in the high-permittivity layer (solid line) for different structures of the primitive cell at a low-contrast PC near zero-width BG. (a) $\alpha =-0.05$; (b) $\alpha =-0.98$, the fraction of the electric field energy is the same on both sides of the BG; and (c) $\alpha =-0.2$. ${\alpha }_{zero}=0$. Other parameters are the same as in Fig. 3. BG regions are shaded.Reuse & Permissions