This paper describes Meep, a popular free implementation of the finite-difference time-domain (FDTD) method for simulating electromagnetism. In particular, we focus on aspects of implementing a full-featured FDTD package that go beyond standard textbook descriptions of the algorithm, or ways in which Meep differs from typical FDTD implementations. These include pervasive interpolation and accurate modeling of subpixel features, advanced signal processing, support for nonlinear materials via Padé approximants, and flexible scripting capabilities. PACS

Forces arising from overlap between the guided waves of parallel, microphotonic waveguides are calculated. Both attractive and repulsive forces, determined by the choice of relative input phase, are found. Using realistic parameters for a silicon-on-insulator material system, we estimate that the forces are large enough to cause observable displacements. Our results illustrate the potential for a broader class of optically tunable microphotonic devices and microstructured artificial materials.

Perturbation theory permits the analytic study of small changes on known solutions, and is especially useful in electromagnetism for understanding weak interactions and imperfections. Standard perturbation-theory techniques, however, have difficulties when applied to Maxwell's equations for small shifts in dielectric interfaces (especially in high-index-contrast, three-dimensional systems) due to the discontinuous field boundary conditions--in fact, the usual methods fail even to predict the lowest-order behavior. By considering a sharp boundary as a limit of anisotropically smoothed systems, we are able to derive a correct first-order perturbation theory and mode-coupling constants, involving only surface integrals of the unperturbed fields over the perturbed interface. In addition, we discuss further considerations that arise for higher-order perturbative methods in electromagnetism.

We present an analytical model and numerical experiments to describe optimal bistable switching in a nonlinear photonic crystal system. It is proved that only three parameters are needed to characterize a bistable switch: the resonant frequency omega(res), the quality factor Q, and parameter kappa that measures nonlinear "feedback strength." A photonic crystal enables the device to operate in single-mode fashion, as if it were effectively one dimensional. This provides optimal control over the input and output and facilitates further large-scale optical integration.

We demonstrate how slow group velocities of light, which are readily achievable in photonic-crystal systems, can dramatically increase the induced phase shifts caused by small changes in the index of refraction. Such increased phase sensitivity may be used to decrease the sizes of many devices, including switches, routers, all-optical logical gates, wavelength converters, and others. At the same time a low group velocity greatly decreases the power requirements needed to operate these devices. We show how these advantages can be used to design switches smaller than 20 m ϫ 200 m in size by using readily available materials and at modest levels of power. With this approach, one could have ϳ10 5 such devices on a surface that is 2 cm ϫ 2 cm, making it an important step towards large-scale all-optical integration.

We prove that an adiabatic theorem generally holds for slow tapers in photonic crystals and other strongly grated waveguides with arbitrary index modulation, exactly as in conventional waveguides. This provides a guaranteed pathway to efficient and broad-bandwidth couplers with, e.g., uniform waveguides. We show that adiabatic transmission can only occur, however, if the operating mode is propagating (nonevanescent) and guided at every point in the taper. Moreover, we demonstrate how straightforward taper designs in photonic crystals can violate these conditions, but that adiabaticity is restored by simple design principles involving only the independent band structures of the intermediate gratings. For these and other analyses, we develop a generalization of the standard coupled-mode theory to handle arbitrary nonuniform gratings via an instantaneous Bloch-mode basis, yielding a continuous set of differential equations for the basis coefficients. We show how one can thereby compute semianalytical reflection and transmission through crystal tapers of almost any length, using only a single pair of modes in the unit cells of uniform gratings. Unlike other numerical methods, our technique becomes more accurate as the taper becomes more gradual, with no significant increase in the computation time or memory. We also include numerical examples comparing to a well-established scattering-matrix method in two dimensions.

Finite-difference time-domain (FDTD) methods suffer from reduced accuracy when modeling discontinuous dielectric materials, due to the inhererent discretization (pixelization). We show that accuracy can be significantly improved by using a subpixel smoothing of the dielectric function, but only if the smoothing scheme is properly designed. We develop such a scheme based on a simple criterion taken from perturbation theory and compare it with other published FDTD smoothing methods. In addition to consistently achieving the smallest errors, our scheme is the only one that attains quadratic convergence with resolution for arbitrarily sloped interfaces. Finally, we discuss additional difficulties that arise for sharp dielectric corners.

We present the light-propagation characteristics of OmniGuide fibers, which guide light by concentric multi-layer dielectric mirrors having the property of omnidirectional reflection. We show how the lowest-loss TE_01 mode can propagate in a single-mode fashion through even large-core fibers, with other modes eliminated asymptotically by their higher losses and poor coupling, analogous to hollow metallic microwave waveguides. Dispersion, radiation leakage, material absorption, nonlinearities, bending, acircularity, and interface roughness are considered with the help of leaky modes and perturbation theory, and both numerical results and general scaling relations are presented. We show that cladding properties such as absorption and nonlinearity are suppressed by many orders of magnitude due to the strong confinement in a hollow core, and other imperfections are tolerable, promising that the properties of silica fibers may be surpassed even when nominally poor materials are employed.