Benchmarking five computational methods for analyzing large photonic crystal membrane cavities Niels Gregersen, Jakob Rosenkrantz de Lasson, Lars Hagedorn Frandsen, Teppo Häyrynen, Andrei Lavrinenko and Jesper Mørk DTU Fotonik, Department of Photonics Engineering Technical University of Denmark Kongens Lyngby, Denmark ngre@fotonik.dtu.dk Oleksiy S. Kim and Olav Breinbjerg DTU Elektro, Department of Electrical Engineering Technical University of Denmark Kongens Lyngby, Denmark Fengwen Wang and Ole Sigmund DTU Mekanik, Department of Mechanical Engineering Technical University of Denmark Kongens Lyngby, Denmark Aliaksandra Ivinskaya ITMO University St. Petersburg, Russia Philipp Gutsche and Sven Burger Zuse Institute Berlin Berlin, Germany Abstract— We benchmark five state-of-the-art computational methods by computing quality factors and resonance wavelengths in photonic crystal membrane L5 and L9 line defect cavities. The convergence of the methods with respect to resolution, degrees of freedom and number of modes is investigated. Convergence is not obtained for some of the methods, indicating that some are more suitable than others for analyzing line defect cavities. Keywords—Photonic crystal; microcavity; line defect cavity; quality factor; numerical simulations I. INTRODUCTION The photonic crystal (PhC) membrane represents a platform for planar integration of components, where cavities and waveguides may play a key role in realizing compact optical components with classical functionality [1] such as switches, lasers [2], and amplifiers or quantum optical functionality [3] such as integrated sources of quantum light. By leaving out a row of holes in an otherwise perfect PhC membrane lattice, a line defect is created in which light may be guided. If the waveguide is terminated at both ends, the finite-length waveguide forms an Ln cavity as illustrated in Fig. 1, where n denotes the length of the cavity. Such Ln cavities support spectrally discrete optical modes, and the fundamental cavity mode profile of an L9 cavity is shown in Fig. 2. Light may be confined to such an Ln cavity for extended periods, as quantified by the quality (Q) factor. For laser applications, the Q factor governs the onset of lasing, and for cavity quantum electrodynamics applications, it governs the onset of strong coupling. The Q factor thus represents a key parameter in the design of a PhC membrane cavity. The combination of the large size of the PhC Ln cavity and the full 3D nature of the geometry makes the calculation of the cavity Q factor an extremely demanding numerical challenge. No matter which numerical method is used, careful convergence checks with respect to the degrees of freedom must be made. Additionally, most numerical simulations methods rely on a closed simulation domain, and here the influence of the boundary conditions requires carefully study. A study of PhC nanobeam cavities using four numerical techniques has previously been reported [4], where cavity frequencies and Q factors were investigated as function of structural parameters. While qualitative agreement between the methods was found, quantitative discrepancies were in some cases as large as an order of magnitude, and estimates for the computational error and the influence of the size of the computational domain were not given. II. OUR COMPUTATIONAL METHODS We employ five different computational methods [5], the finite-difference time-domain (FDTD) technique, the finite- difference frequency-domain (FDFD) technique, the finite- element method (FEM), the surface integral equation (SIE) approach and the Fourier modal method (FMM), to compute the cavity Q factor for the Ln cavities. Three variations of the FEM method are considered, eigenvalue (FEM1) and scattering (FEM2) analysis using the commercial JCMWAVE software package and eigenvalue (FEM3) analysis using the COMSOL package. In this work, we focus on two structures, a low-Q L5 cavity and a high-Q L9 cavity. The boundary of the structure is chosen such the boundary cylinders are half circles, and the surrounding air region is, in principle, of infinite extent. We then determine the wavelength and Q factor of the fundamental Fig. 1. The geometry and the optical field |Ey| 2 profile for the L9 cavity mode.