Electromagnetic modes of a disordered photonic crystal

Abstract

We present a systematic numerical approach to compute the eigenmodes and the related eigenfrequencies of a disordered photonic crystal, characterized by small fluctuations of the otherwise periodic dielectric profile. The field eigenmodes are expanded on the basis of Bloch modes of the corresponding periodic structure, and the resulting eigenvalue problem is diagonalized on a truncated basis including a finite number of Bloch bands. The Bloch-mode expansion is very effective for modeling modes of very extended disordered structures in a given frequency range, as only spectrally close bands must be included in the basis set. The convergence can be easily verified by repeating the diagonalization for an increased band set. As illustrations, we apply the method to the W1 line-defect waveguide and to the L3 coupled-cavity waveguide, both based on a photonic crystal slab with a triangular lattice of circular holes. We compute and characterize the eigenfrequencies, spatial field profiles, and radiation loss rates of the localized modes induced by disorder. For the W1 waveguide, we demonstrate that radiation losses, at the bottom of the spatially even guided band, are determined by a small hybridization with Bloch modes of the spatially odd band, induced by disorder in spite of their frequency separation. The Bloch-mode expansion method has a very broad range of applications: It can be also used to accurately compute the modes of structures with systematic modulations of the periodic dielectric constant, as in several designs of high-Q cavities.

Radiative coupling of quantum dots in photonic crystal structures

“…Apart from small variations of the field profiles, 105 that should only marginally influence the magnitude of the coupling between one QD and a photon mode, the main effect of disorder is the localization of light in long PHC structures. 86,[119][120][121] For the line defects studied in this work in particular, localization is known to compete with the ability of the waveguide to support slow-light propagation, 122,123 and thus with the enhancement of radiative effects expected in these structures. More specifically, when approaching the band edge in a W1 waveguide, the localization length becomes shorter than the decay length related to extrinsic radiation losses.…”

“…Then, the assumption of approximately constant Q's is realistic, as seen from microscopic modeling of extrinsic disorder-induced losses. 86,113 Polariton modes originating from antisymmetric photon modes are essentially uncoupled and are not displayed (although they were still included in the computation). The coupling constants of each of the dots to each of the symmetric modes varies very little, and ish|g…”

“…The last requirement of the problem is the knowledge of the modes of the PHC structure, i.e., the set of orthonormal functions {Q m (r)} and their corresponding eigenfrequencies ω m . Here, PHC modes are computed using the Bloch-mode expansion method, 86 which consists of expanding the modes on the basis of the Bloch modes of a regular waveguide. These latter were in turn computed using an expansion over the guided modes of a uniform dielectric slab.…”

“…In particular, we frame the underlying Maxwell equations into an eigenvalue problem, describing the polariton modes of the system in analogy with the polariton formalisms for a bulk semiconductor, 84 for quantum wells, 85 and for QDs in an unstructured photonic environment. 55,57 For the computation of the photonic modes, the Bloch-mode expansion method is employed, 86 although any other method which provides reliable field profiles (e.g., finite-element method (FEM), finite-difference time-domain (FDTD)) can also be used. Even though the modeling of radiative effects in the presence of fabrication disorder lies beyond the scope of the present work, the Bloch-mode expansion is particularly well suited for treating large, disordered photonic structures 86,87 and was thus an obvious choice in view of a future extension to disordered PHCs.…”

“…55,57 For the computation of the photonic modes, the Bloch-mode expansion method is employed, 86 although any other method which provides reliable field profiles (e.g., finite-element method (FEM), finite-difference time-domain (FDTD)) can also be used. Even though the modeling of radiative effects in the presence of fabrication disorder lies beyond the scope of the present work, the Bloch-mode expansion is particularly well suited for treating large, disordered photonic structures 86,87 and was thus an obvious choice in view of a future extension to disordered PHCs. We apply the formalism to Ln cavities and W 1 waveguides based on a PHC slab.…”