Opening up complete photonic bandgaps in three-dimensional photonic crystals consisting of biaxial dielectric spheres

Shiyang Liu and Zhifang Lin

Phys. Rev. E 73, 066609 – Published 8 June 2006

Abstract

In this paper, the scattering matrix of sphere with dielectric biaxial anisotropy is obtained exactly within the framework of the extended Mie theory. By incorporating the scattering matrix into the multiple scattering method, we study theoretically the photonic band structure of three dimensional photonic crystals consisting of biaxial dielectric spheres. Our results demonstrate that complete photonic bandgaps can be found in both fcc and sc lattice structures, which are absent in photonic crystals composed of isotropic dielectric spheres. Moreover we have compared our results with those coming from the photonic crystals consisting of the uniaxially birefringent dielectric spheres. It is found that because of the enhancement of the anisotropy, the degeneracy is lifted further, resulting in two neighboring photonic bandgaps, the lower one is complete while the upper one is partial existing only in some special regions of the first Brillouin zone.

Received 17 December 2005

DOI:https://doi.org/10.1103/PhysRevE.73.066609

Authors & Affiliations

Shiyang Liu^*

Surface Physics Laboratory, Fudan University, Shanghai 200433, P. R. China

Zhifang Lin^†

Department of Physics, Fudan University, Shanghai 200433, P. R. China

^*Electronic address: 041019018@fudan.edu.cn
^†Electronic address: phlin@fudan.ac.cn

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Vol. 73, Iss. 6 — June 2006

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Figure 1
Calculated photonic band structures of a PC consisting of biaxial dielectric spheres in fcc lattice with permittivity tensor $ϵ_{s} = 35$ , $ϵ_{1} = 0.65$ , $ϵ_{2} = 0.75$ , $ϵ_{3} = 1.0$ . The extraordinary axis, corresponding to the principal axis $ϵ_{3}$ with $ϵ_{3} = 1.0$ , is oriented along the [111] direction and the two ordinary axes $ϵ_{1}$ and $ϵ_{2}$ are along the $[11 \bar{2}]$ and $[\bar{1} 10]$ directions, respectively. The filling fraction is $f = 0.25$ . The photonic band structures of this PC in $24$ partial regions (one-half of the first Brillouin zone) are calculated. The above eight diagrams correspond to eight partial regions respectively in the first Brillouin zone with the high symmetry points: (a) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (0, 1, 0)$ , $U = (2 π ∕ a) (\frac{1}{4}, 1, \frac{1}{4})$ , $L = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ , $W = (2 π ∕ a) (\frac{1}{2}, 1, 0)$ , $K = (2 π ∕ a) (\frac{3}{4}, \frac{3}{4}, 0)$ ; (b) the same as (a) except $W = (2 π ∕ a) (0, 1, \frac{1}{2})$ , $K = (2 π ∕ a) (0, \frac{3}{4}, \frac{3}{4})$ ; (c) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (0, 0, 1)$ , $U = (2 π ∕ a) (\frac{1}{4}, \frac{1}{4}, 1)$ , $L = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ , $W = (2 π ∕ a) (0, \frac{1}{2}, 1)$ , $K = (2 π ∕ a) (0, \frac{3}{4}, \frac{3}{4})$ ; (d) the same as (c) except $W = (2 π ∕ a) (\frac{1}{2}, 0, 1)$ , $K = (2 π ∕ a) (\frac{3}{4}, 0, \frac{3}{4})$ ; (e) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (0, 0, - 1)$ , $U = (2 π ∕ a) (\frac{1}{4}, \frac{1}{4}, - 1)$ , $L = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, - \frac{1}{2})$ , $W = (2 π ∕ a) (0, \frac{1}{2}, - 1)$ , $K = (2 π ∕ a) (0, \frac{3}{4}, - \frac{3}{4})$ ; (f) the same as (e) except $W = (2 π ∕ a) (\frac{1}{2}, 0, - 1)$ , $K = (2 π ∕ a) (\frac{3}{4}, 0, - \frac{3}{4})$ ; (g) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (1, 0, 0)$ , $U = (2 π ∕ a) (1, \frac{1}{4}, - \frac{1}{4})$ , $L = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, - \frac{1}{2})$ , $W = (2 π ∕ a) (1, 0, - \frac{1}{2})$ , $K = (2 π ∕ a) (\frac{3}{4}, 0, - \frac{3}{4})$ ; (h) the same as (g) except $W = (2 π ∕ a) (1, \frac{1}{2}, 0)$ , $K = (2 π ∕ a) (\frac{3}{4}, \frac{3}{4}, 0)$ .Reuse & Permissions
Figure 2
The gap-to-midgap ratio of the complete band gap vs filling fraction from the PC in fcc lattice structure with $ϵ_{s} = 35$ , $ϵ_{1} = 0.65$ , $ϵ_{2} = 0.75$ and $ϵ_{3} = 1.0$ . The extraordinary axis, corresponding to the principal axis $ϵ_{3}$ with $ϵ_{3} = 1.0$ , is oriented along the [111] direction and the two ordinary axes $ϵ_{1}$ and $ϵ_{2}$ are along the $[11 \bar{2}]$ and $[\bar{1} 10]$ directions respectively.Reuse & Permissions
Figure 3
The gap-to-midgap ratios of the partial PBGs as a function of filling fraction from the PC in fcc lattice structure composed of biaxial dielectric spheres with $ϵ_{s} = 35$ , $ϵ_{1} = 0.65$ , $ϵ_{2} = 0.75$ , $ϵ_{3} = 1.0$ . The extraordinary axis, corresponding to the principal axis $ϵ_{3}$ with $ϵ_{3} = 1.0$ , is oriented along the [111] direction and the two ordinary axes $ϵ_{1}$ and $ϵ_{2}$ are along the $[11 \bar{2}]$ and $[\bar{1} 10]$ directions respectively. The above two diagrams correspond to two partial regions of the first Brillouin zone with high symmetry points (a) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (0, 0, - 1)$ , $U = (2 π ∕ a) (\frac{1}{4}, \frac{1}{4}, - 1)$ , $L = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, - \frac{1}{2})$ , $W = (2 π ∕ a) (0, \frac{1}{2}, - 1)$ , $K = (2 π ∕ a) (0, \frac{3}{4}, - \frac{3}{4})$ ; (b) the same as (a) except $W = (2 π ∕ a) (\frac{1}{2}, 0, - 1)$ , $K = (2 π ∕ a) (\frac{3}{4}, 0, - \frac{3}{4})$ .Reuse & Permissions
Figure 4
Calculated photonic band structures of a PC consisting of biaxial dielectric spheres in sc lattice with permittivity tensor $ϵ_{s} = 35$ , $ϵ_{1} = 0.60$ , $ϵ_{2} = 0.65$ , $ϵ_{3} = 1.0$ . The extraordinary axis, corresponding to the principal axis $ϵ_{3}$ with $ϵ_{3} = 1.0$ , is oriented along the [111] direction and the two ordinary axes $ϵ_{1}$ and $ϵ_{2}$ are along the $[11 \bar{2}]$ and $[\bar{1} 10]$ directions respectively. The filling fraction is $f = 0.23$ . The photonic band structures of this PC in $24$ partial regions (one-half of the first Brillouin zone) are plotted. The above eight diagrams correspond to eight partial regions, respectively, in the first Brillouin zone with the high symmetry points: (a) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (0, \frac{1}{2}, 0)$ , $R = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ , $M = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, 0)$ ; (b) the same as (a) except $M = (2 π ∕ a) (0, \frac{1}{2}, \frac{1}{2})$ ; (c) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (0, 0, \frac{1}{2})$ , $R = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ , $M = (2 π ∕ a) (0, \frac{1}{2}, \frac{1}{2})$ ; (d) the same as (c) except $M = (2 π ∕ a) (\frac{1}{2}, 0, \frac{1}{2})$ ; (e) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (0, 0, - \frac{1}{2})$ , $R = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, - \frac{1}{2})$ , $M = (2 π ∕ a) (0, \frac{1}{2}, - \frac{1}{2})$ ; (f) the same as (e) except $M = (2 π ∕ a) (\frac{1}{2}, 0, - \frac{1}{2})$ ; (g) $Γ = (0, 0, 0)$ , $X = (2 π ∕ a) (\frac{1}{2}, 0, 0)$ , $R = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, - \frac{1}{2})$ , $M = (2 π ∕ a) (\frac{1}{2}, 0, - \frac{1}{2})$ ; (h) the same as (g) except $M = (2 π ∕ a) (\frac{1}{2}, \frac{1}{2}, 0)$ .Reuse & Permissions

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