Cartesian light: Unconventional propagation of light in a three-dimensional superlattice of coupled cavities within a three-dimensional photonic band gap

Abstract

We explore the unconventional propagation of light in a three-dimensional (3D) superlattice of coupled resonant cavities in a 3D photonic band-gap crystal. Such a 3D cavity superlattice is the photonic analog of the Anderson model for spins and electrons in the limit of zero disorder. Using the plane-wave expansion method, we calculate the dispersion relations of the 3D cavity superlattice with the cubic inverse woodpile structure that reveal five coupled-cavity bands, typical of quadrupole-like resonances. For three out of five bands, we observe that the dispersion bandwidth is significantly larger in the (k x , k z)-diagonal directions than in other directions. To explain the directionality of the dispersion bandwidth, we employ the tight-binding method from which we derive coupling coefficients in three dimensions. For all converged coupled-cavity bands, we find that light hops predominantly in a few high-symmetry directions including the Cartesian (x, y, z) directions, therefore we propose the name "Cartesian light." Such 3D Cartesian hopping of light in a band gap yields propagation as superlattice Bloch modes that differ fundamentally from the conventional 3D spatially extended Bloch wave propagation in crystals, from light tunneling through a band gap, from coupled-resonator optical waveguiding, and also from light diffusing at the edge of a gap.

Three-dimensional photonic band gap cavity with finite support: Enhanced energy density and optical absorption

“…Firstly, the consistent observation of five resonances by three independent studies (Refs. [50,54] and the present work) strongly suggests that a cavity in an inverse woodpile photonic band gap crystal has eigenstates with quadrupole symmetry, Figure 14: Fano resonances below the 3D band gap for a 3D inverse woodpile photonic crystal with a cavity. The red squares in (a) and the green circles in (b) are zoomed-in reflectivity spectra calculated at normal incidence for s and p polarizations, respectively.…”

Computational nanophotonics in 3D

“…The discontinuous Galerkin finite element method (DGFEM) [127][128][129][130][131][132][133] is mathematically proven to be much better suited for hp-adaptation than the conforming FEM. The DG method allows to accurately capture singularities at dielectric corners (that occur, for instance, in resonant cavities [87,90]) by locally increasing the spatial resolution (hp-refinement). Moreover, the DG method treats elements individually due to the use of element-wise discontinuous basis functions and hence is ideally suited for adaptation and, since it results in a (block)-diagonal mass matrix, also for time-stepping [104].…”

“…The propagation of light in such a 3D cavity superlattice is analogous to electronic transport in an impurity band in a semiconductor [43,51,89]. The 3D cavity superlattice has superlattice Bloch modes where photons hop between cavities [90].…”

Controlled light propagation in random, periodic, and superperiodic silicon nanophotonic materials

“…In 3D superlattices, cavities are periodically repeated with a longer lattice constant in the photonic crystal, thereby making the structure superperiodic. Recently, theoretical work in our research group has predicted the occurrence of intricate "Cartesian light" in superperiodic medium, wherein light propagates by hopping only in a few high symmetry directions in space with different coupling strengths [97]. Such propagation differs fundamentally from the conventional spatially-extended Bloch wave propagation outside the photonic gap 2 .…”

“…When the radii of the pair of defect pores are tuned to r ′ = r/2, light is maximally confined inside the cavity [115]. When multiple cavities are repeated periodically in the crystal, coupling between the cavities results in a 3D cavity superlattice that sustains 'Cartesian light' [97], which will be explained in detail in chapter 5. In presence of cavities, additional states are created inside the original band gap of the underlying 3D band gap crystal.…”