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

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Although perfectly matched layers (PMLs) have been widely used to truncate numerical simulations of electromagnetism and other wave equations, we point out important cases in which a PML fails to be reflectionless even in the limit of infinite resolution. In particular, the underlying coordinate-stretching idea behind PML breaks down in photonic crystals and in other structures where the material is not an analytic function in the direction perpendicular to the boundary, leading to substantial reflections. The alternative is an adiabatic absorber, in which reflections are made negligible by gradually increasing the material absorption at the boundaries, similar to a common strategy to combat discretization reflections in PMLs. We demonstrate the fundamental connection between such reflections and the smoothness of the absorption profile via coupled-mode theory, and show how to obtain higher-order and even exponential vanishing of the reflection with absorber thickness (although further work remains in optimizing the constant factor).

We present a general framework for the design of thin-film photovoltaics based on a partiallydisordered photonic crystal that has both enhanced absorption for light trapping and reduced sensitivity to the angle and polarization of incident radiation. The absorption characteristics of different lattice structures are investigated as an initial periodic structure is gradually perturbed. We find that an optimal amount of disorder controllably introduced into a multi-lattice photonic crystal causes the characteristic narrow-band, resonant peaks to be broadened resulting in a device with enhanced and robust performance ideal for typical operating conditions of photovoltaic applications.

We present a unified density-based topology-optimization framework that yields integrated photonic designs optimized for manufacturing constraints including all those of commercial semiconductor foundries. We introduce a new method to impose minimum-area and minimum-enclosed-area constraints, and simultaneously adapt previous techniques for minimum linewidth, linespacing, and curvature, all of which are implemented without any additional re-parameterizations. Furthermore, we show how differentiable morphological transforms can be used to produce devices that are robust to over/under-etching while also satisfying manufacturing constraints. We demonstrate our methodology by designing three broadband silicon-photonics devices for nine different foundry-constraint combinations.

We investigate the design of taper structures for coupling to slow-light modes of various photonic-crystal waveguides while taking into account parameter uncertainties inherent in practical fabrication. Our short-length (11 periods) robust tapers designed for ? = 1.55?m and a slow-light group velocity of c/34 have a total loss of < 20 dB even in the presence of nanometer-scale surface roughness, which outperform the corresponding non-robust designs by an order of magnitude. We discover a posteriori that the robust designs have smooth profiles that can be parameterized by a few-term (intrinsically smooth) sine series which helps the optimization to further boost the performance slightly. We ground these numerical results in an analytical foundation by deriving the scaling relationships between taper length, taper smoothness, and group velocity with the help of an exact equivalence with Fourier analysis.

Finite-difference time-domain methods suffer from reduced accuracy when discretizing discontinuous materials. We previously showed that accuracy can be significantly improved by using subpixel smoothing of the isotropic dielectric function, but only if the smoothing scheme is properly designed. Using recent developments in perturbation theory that were applied to spectral methods, we extend this idea to anisotropic media and demonstrate that the generalized smoothing consistently reduces the errors and even attains second-order convergence with resolution. , and replaces our previous heuristic proposal for anisotropic-material interfaces [5]. Ordinarily, the presence of discontinuous material interfaces degrades the accuracy of FDTD to first-order ͓O͑⌬x͔͒ from the usual second-order ͓O͑⌬x 2 ͔͒ accuracy [6], but our work demonstrates how an appropriate choice of subpixel smoothing can both restore second-order asymptotic accuracy and give the lowest errors compared with competing schemes even at modest resolutions. Subpixel smoothing has an additional benefit: it allows the simulation to respond continuously to changes in the geometry, such as during optimization or parameter studies, rather than changing in discontinuous jumps as interfaces cross pixel boundaries. This technique additionally yields much smoother convergence of the error with resolution, which makes it easier to evaluate the accuracy and enables the possibility of extrapolation to gain another order of accuracy [4]. The ability to handle anisotropic materials is becoming increasingly important via the use of anisotropic materials to represent arbitrary coordinate transformations in Maxwell's equations [7], most prominently to design cloaking metamaterials [8]. Our smoothing scheme requires preprocessing of the materials and does not otherwise modify the FDTD algorithm. It is therefore particularly simple to implement (free software is available [9]).Our basic approach, as described previously [2,3], is to smooth the structure to eliminate the discontinuity before discretizing, but because the smoothing itself changes the geometry we use first-order perturbation theory to select a smoothing with zero firstorder effect. For isotropic materials, this approach made rigorous a smoothing scheme that had previously been proposed heuristically [10][11][12] and explained its second-order accuracy [3]. Advances in perturbation theory have enabled us to extend this scheme to interfaces between anisotropic materials, initially for a plane-wave method [2]. Here, we adapt the technique to FDTD, combined with a recent FDTD scheme with improved stability for anisotropic media [4]. Although this Letter focuses on the case of anisotropic electric permittivity , exactly the same smoothing and discretization schemes apply to magnetic permeabilities µ owing to the equivalence in Maxwell's equations under interchange of /µ and E / H.We define an interface-relative coordinate frame as in Fig. 1, so that the first component "1" is the direction normal to the interface. ...

We present a photonics topology optimization (TO) package capable of addressing a wide range of practical photonics design problems, incorporating robustness and manufacturing constraints, which can scale to large devices and massive parallelism. We employ a hybrid algorithm that builds on a mature time-domain (FDTD) package Meep to simultaneously solve multiple frequency-domain TO problems over a broad bandwidth. This time/frequency-domain approach is enhanced by new filter-design sources for the gradient calculation and new material-interpolation methods for optimizing dispersive media, as well as by multiple forms of computational parallelism. The package is available as free/open-source software with extensive tutorials and multi-platform support.