Bloch Modes and Evanescent Modes of Photonic Crystals: Weak Form Solutions Based on Accurate Interface Triangulation

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 ζ. 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.

Coexistence of both gyroid chiralities in individual butterfly wing scales of Callophrys rubi

“…The single-gyroid material (of a given fixed handedness) has been clearly shown to discriminate between LH and RH circular polarized light (13,14,38,41,42). Given the prevalence of the LH enantiomer of the single gyroid and the frequent occurrence of the <001> inclination [being the sole direction with strong circular dichroism (12,23)], the wing scales of C. rubi fulfill the prerequisites for the existence of an optical circular polarization signal. However, spectral measurements on the wings of C. rubi (and T. imperialis, another gyroid-forming butterfly species) have failed to record any discrimination between left and right circularly polarized light (12,20).…”

“…It is an essential determinant of color uniformity, because the photonic properties of the single gyroid strongly depend on the crystal orientation (12,23). Yoshioka et al (9) observed a highly textured crystal orientation within scales of the butterfly Parides sesostris.…”

Optical Chirality Flux as a Useful Far-Field Probe of Chiral Near Fields

“…To maximize the effect, the near field should first be characterized through an appropriate far-field quantity. Typically, this has been done with circular dichroism (CD) spectroscopy, which records the difference in the optical extinction from a chiral object when it is illuminated with left-and right-handed circularly polarized light (for alternative methods see [21,22]). For most biomolecules, the differential absorption dominates the CD signal and provides information about the chiral arrangement of the molecular components.…”

Sheet-like chiro-optical material designs based C(Y) surfaces

“…?, 12 In this latter approach (described in detail in, 12, 13 see also 14,15 ), coupling amplitudes of an incident circularly-polarized plane wave with the crystal Bloch modes are computed in the crystal cut-off plane perpendicular to the wave-vector of the incoming wave. A circular dichroism index C ∈ [−1, 1] is derived that measures the dichroism strength, with C = 0 corresponding to no dichroism and C = ±1 to maximal dichroism.…”

“…This article uses the exact definitions of ref. 13 This method is a coarse approximation of a more rigorously defined scattering method; see reference 12,13 for details of the scattering method and for a numeric implementation. Figures 2 and 3 show the circular-dichroism optical response of a double network structure (Fig.…”