Abstract
We propose a new approach to calculate the complex photonic band structure, both purely dispersive and evanescent Bloch modes of a finite range, of arbitrary three-dimensional photonic crystals. Our method, based on a well-established plane wave expansion and the weak form solution of Maxwell’s equations, computes the Fourier components of periodic structures composed of distinct homogeneous material domains from a triangulated mesh representation of the inter-material interfaces; this allows substantially more accurate representations of the geometry of complex photonic crystals than the conventional representation by a cubic voxel grid. Our method works for general two-phase composite materials, consisting of bi-anisotropic materials with tensor-valued dielectric and magnetic permittivities ε and μ and coupling matrices ς. We demonstrate for the Bragg mirror and a simple cubic crystal closely related to the Kelvin foam that relatively small numbers of Fourier components are sufficient to yield good convergence of the eigenvalues, making this method viable, despite its computational complexity. As an application, we use the single gyroid crystal to demonstrate that the consideration of both conventional and evanescent Bloch modes is necessary to predict the key features of the reflectance spectrum by analysis of the band structure, in particular for light incident along the cubic [111] direction.
Highlights
A photonic crystal (PhC) is a composite material with a periodic and symmetric spatial arrangement of at least two components with different macroscopic electromagnetic properties, such as the dielectric constant ε and, more generally, the magnetic permittivity μ and the coupling constant ζ [3]
The existence of a photonic band gap, i.e., a frequency range for which no Bloch modes exist in the infinite PhC, is commonly interpreted as a stop band, i.e., a frequency interval for which no light can penetrate through the photonic crystal and is, reflected
The coefficient largely depends on the quality of geometrical representation that is approximated with linear accuracy in the case of MPB and with negligible error in our case and on the field intensities of the respective mode near the surface where the plane wave expansion produces a large deviation from the true solution
Summary
A photonic crystal (PhC) is a composite material with a periodic and symmetric spatial arrangement of at least two components with different macroscopic electromagnetic properties, such as the dielectric constant ε (or only its real part in the original definition [1,2]) and, more generally, the magnetic permittivity μ and the coupling constant ζ [3]. Simple approaches to evaluate the degree of coupling between an incoming plane wave and the Bloch modes of a PhC suffice in many situations to correlate features of the band structure to the transmission or reflectance spectra [4], e.g., for light incident onto a semi-infinite single gyroid or a four-srs PhC along the r100s or r110s lattice direction [5]. We use a similar approach as in [13] and solve the bulk eigenvalue equation in Fourier space to obtain the generalized Bloch modes, including those that are evanescent This earlier work [13] did not mention how the Fourier transforms are performed while presumably having the same memory limitation (without mentioning this fact explicitly) due to matrix inversion (see Section 3.2), as the authors show a convergence diagram for a very small number of plane waves N ă 1500 in a two-dimensional case only. (b) a commonly-used representation (e.g., in the MIT packages MEEP [28] or MBP [17]) is the voxelised "LEGOr -like" representation, where space is represented by a regular voxel grid with each voxel being either solid or void; this representation always leads to stair-case errors; see, in particular, the Appendix of [18]; (c) a representation of the geometry by a mesh (or triangulation) representing the interface between solid and void yields more accurate geometric representations
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