Omnidirectional two-dimensional photonic crystal band gap structures

Omnidirectional (3D) band gaps can evolve in $2\frac{1}{2}$ honeycomb and squared photonic crystal lattices and also in hexagonal crystals [Li and Xia, Phys. Rev. B 64, 153 (2001)]. This study presents a study of the according full parameter space. The calculation of omnidirectional photonic bands is based on the plane wave method for a two-dimensional photonic crystal with an off-plane wave vector $k_z=k_0\sin \left(\theta \right)$. The results are fully respecting the parameters index of refraction, the angle of incidence, as well as the structure radii. Omnidirectional band gaps are found for radii between $R=0.34$ and $R=0.5$, starting with index values from $3.0$ and show maximum relative band gap sizes ranging from $2.7\%$ up to $10\%$.

Figure 1

Schematically measurement setup for transmission and reflection as a function of frequency for various angles of incidence $\theta$. The planes characterize constant phases of plane waves which incident from air into the invariant photonic crystal along the $z$-axis with an off-plane wave vector $k_z=k_0\sin \left(\theta \right)$.

Figure 2

Band diagrams as a function of off-plane incidence angle $\theta$ for a squared lattice with $n_1=4$, $n_2=1$ and $R=0.48a$ calculated by Eq. (2). The lattice vectors are ${\mathbf{a}}_1=a[1;0{]}^T$, ${\mathbf{a}}_2=a[0;1{]}^T$ with the high symmetry points of the first Brillouin zone $\mathbf{\Gamma }=[0;0{]}^T$, $\mathbf{X}=\pi ∕a[0;1{]}^T$ and $\mathbf{M}=\pi ∕a[1;1{]}^T$. A common band gap is shown by the gray bar for left: $\theta =0^{°}$, center: $\theta ={45}^{°}$ and right: $\theta ={90}^{°}$.

Figure 3

Band edges of bands 5 and 6 as a function of off-plane incidence $\theta$ for a squared lattice (Fig. 2) with a resulting omnidirectional PBG.

Figure 4

Absolute band gap for $\theta =0^{°}$ and for off-plane angle $\theta ={90}^{°}$ with the resultant omnidirectional PBG as a function of refraction index $n_1$ for a squared lattice with $n_2=1$ and $R=0.48a$.

Figure 5

Relative omnidirectional PBG as a function of normalized radius $R∕a$ for a squared lattice with $n_1=4$ and $n_2=1$.

Figure 6

Relative omnidirectional PBG between bands 5 and 6 for a squared lattice as a function of refraction index $n_1$ and normalized radius $R∕a$ with $n_2=1$.

Figure 7

Omnidirectional relative PBG between bands 3 and 4 for a hexagonal lattice as a function of refraction index $n_1$ and normalized radius $R∕a$ with $n_2=1$.

Figure 8

Omnidirectional relative band gap of honeycomb lattice as a function of refraction index $n_2$ and normalized radius $R∕a$, with $n_1=1$. The PBG situated between bands 11 and 12.

Figure 9

Omnidirectional relative band gap of modified squared lattice as a function of normalized radii $R_1∕a$ and $R_2∕a$ with $n_1=4$ and $n_2=1$. The lattice vectors are ${\mathbf{a}}_1=a[1;0{]}^T$, ${\mathbf{a}}_2=a[0;1{]}^T$ with the high symmetry points of the first Brillouin zone $\mathbf{\Gamma }=[0;0{]}^T$, $\mathbf{X}=\pi ∕a[0;1{]}^T$, and $\mathbf{M}=\pi ∕a[1;1{]}^T$.

Figure 10

Omnidirectional relative band gap of modified squared lattice as a function of refraction index $n_1$ and normalized radius $R_1=0.495a$, $R_2=0.055a$, and $n_2=1$.