Disordered photonic crystals understood by a perturbation formalism

Abstract

Photonic band gaps in disordered two-dimensional photonic crystals are investigated for two typical types of randomness: cylinder site displacements ͑site randomness͒ and cylinder radius variations ͑size randomness͒. The plane-wave expansion method with a supercell technique is applied to calculate the density-of-states ͑DOS͒ for the disordered photonic crystals. In particular, numerical simulations on the DOS for square and triangular lattices of dielectric cylinders in air with the E-polarization mode show that photonic band gaps are far more sensitive to disorders with a size randomness than with a site randomness. The first and second band gaps both reduce very little even for a site randomness of a strength as large as half the cylinder radius, yet they reduce more than one-half for a size randomness of a strength about one-third the cylinder radius. This substantial contrast can be understood by the analysis of the electromagnetic fields in disordered crystals. Based on such a field analysis, a perturbation formalism is proposed for disordered crystals and it accords well with the DOS calculations for a site randomness of even a moderate strength. At very weak size randomness, the perturbation method also works well to some extent. Such a simple perturbative analysis should provide a systematic way to understand various disordered photonic crystals qualitatively and even semiquantitatively.

Fragility of photonic band gaps in inverse-opal photonic crystals

“…In fact, recent work on two-dimensional ͑2D͒ disordered photonic crystal 17 demonstrated that the higher band gap is far more sensitive to the site and size randomness than the ground band gap. Although this is apparent from a simple physical argument, our interest here is to investigate how fragile the gap in inverse-opal crystal could be in the presence of disorder.…”

“…Similar behavior was found in 2D cases. 17 It can be argued qualitatively that varying the size of the sphere means changing the filling fraction, while the fraction does not change when displacing spheres from lattice sites. Thus, the size randomness reduces the gap size more significantly.…”

“…15,16 The convergence for the ten lowest bands can be made better than 0.5% by adopting 343 plane waves. For a disordered crystal, a supercell technique [17][18][19] is employed with a cubic supercell composed of eight conventional fcc unit cells with 32 spheres and an expansion of 2197 plane waves, where the convergence is better than 1.0%. We have compared the results with those from a supercell with four conventional unit cells and found the same behavior.…”

Dispersive elastodynamics of 1D banded materials and structures: Design

“…Several studies have been done on the effects of disorder on band-gap size and density of states (DOS) of photonic crystals as well as on the transmission in these crystals (see, for example, Refs. [25][26][27]). A common approach is to introduce a random disorder on the location or geometric features of the inclusions (such as cylinders or spheres) across several cells, and to subsequently solve for the band structure, DOS, or transmission performance, of a "supercell" that comprises the perturbed cells.…”

High-Q Defect-Free 2D Photonic Crystal Cavity from Random Localised Disorder

“…PhC cavities and the associated modal confinement can be achieved by introducing a defect in the periodic lattice such as removing a hole from a 2D lattice of airholes, which leads to a localized state in the photonic bandgap. An alternative is a defect-free PhC cavity, i.e., a cavity in which none of the air holes is removed from the periodic lattice [16][17][18][19]. Rather, such cavities result in PhCs made of photosensitive chalcogenide glass [20], by the introduction of nanodiamonds into airholes [21], and in PhCs with airholes with graded radii [16,22].…”