Contour-path effective permittivities for the two-dimensional finite-difference time-domain method

Finite-difference Time-domain (FDTD) Optical Simulations: A Primer for the Life Sciences and Bio-Inspired Engineering

Light influences most ecosystems on earth, from sun-dappled forests to bioluminescent creatures in the ocean deep. Biologists have long studied nano- and micro-scale organismal adaptations to manipulate light using ever-more sophisticated microscopy, spectroscopy, and other analytical equipment. In combination with experimental tools, simulations of light interacting with objects can help researchers determine the impact of observed structures and explore how variations affect optical function. In particular, the finite-difference time-domain (FDTD) method is widely used throughout the nanophotonics community to efficiently simulate light interacting with a variety of materials and optical devices. More recently, FDTD has been used to characterize optical adaptations in nature, such as camouflage in fish and other organisms, colors in sexually-selected birds and spiders, and photosynthetic efficiency in plants. FDTD is also common in bioengineering, as the design of biologically-inspired engineered structures can be guided and optimized through FDTD simulations. Parameter sweeps are a particularly useful application of FDTD, which allows researchers to explore a range of variables and modifications in natural and synthetic systems (e.g., to investigate the optical effects of changing the sizes, shape, or refractive indices of a structure). Here, we review the use of FDTD simulations in biology and present a brief methods primer tailored for life scientists, with a focus on the commercially available software Lumerical FDTD. We give special attention to whether FDTD is the right tool to use, how experimental techniques are used to acquire and import the structures of interest, and how their optical properties such as refractive index and absorption are obtained. This primer is intended to help researchers understand FDTD, implement the method to model optical effects, and learn about the benefits and limitations of this tool. Altogether, FDTD is well-suited to (i) characterize optical adaptations and (ii) provide mechanistic explanations; by doing so, it helps (iii) make conclusions about evolutionary theory and (iv) inspire new technologies based on natural structures.

Review and accuracy comparison of various permittivity-averaging schemes for material discontinuities in the two-dimensional FDFD method: implementation using efficient computer graphics techniques

Several known and widely used averaging techniques aiming to improve the accuracy of the two-dimensional finite-difference frequency-domain (FDFD) method, in the presence of material discontinuities, are reviewed, numerically tested, and compared with respect to their accuracies. Furthermore, all averaging techniques are rigorously and efficiently implemented using the Supercover Digital Differential Analyzer algorithm and a modified Liang-Barsky algorithm suitably adapted from computer graphics applications. The FDFD with Gaussian blurring; the FDFD with volume-polarized effective permittivity; the FDFD with volume-polarized effective permittivity on shifted cells; and the FDFD with anisotropic smoothing [FDFD (AS)] are compared with respect to their accuracies (for both TE and TM polarization), in the case of scattering by an infinite homogeneous cylinder (for which analytical solution exists) comprising a lossless dielectric, a high-index, low-loss dielectric, or a metal. Sample plots of the relative errors are presented for various field components. Absolute error norms (L₂ and L_∞) are also presented for both polarizations and for two grid-cell sizes for quantitative comparisons. The results show that the FDFD (AS) prevails in accuracy mainly because it satisfies the boundary conditions at the cylinder's boundary the best. However, for the high-index dielectrics and metals, even the FDFD without any averaging gives very good results for the field components parallel to the uniformity direction. However, the FDFD (AS) is always more accurate when the in-plane field components are sought.

On the convergence and accuracy of the FDTD method for nanoplasmonics

Use of the Finite-Difference Time-Domain (FDTD) method to model nanoplasmonic structures continues to rise - more than 2700 papers have been published in 2014 on FDTD simulations of surface plasmons. However, a comprehensive study on the convergence and accuracy of the method for nanoplasmonic structures has yet to be reported. Although the method may be well-established in other areas of electromagnetics, the peculiarities of nanoplasmonic problems are such that a targeted study on convergence and accuracy is required. The availability of a high-performance computing system (a massively parallel IBM Blue Gene/Q) allows us to do this for the first time. We consider gold and silver at optical wavelengths along with three "standard" nanoplasmonic structures: a metal sphere, a metal dipole antenna and a metal bowtie antenna - for the first structure comparisons with the analytical extinction, scattering, and absorption coefficients based on Mie theory are possible. We consider different ways to set-up the simulation domain, we vary the mesh size to very small dimensions, we compare the simple Drude model with the Drude model augmented with two critical points correction, we compare single-precision to double-precision arithmetic, and we compare two staircase meshing techniques, per-component and uniform. We find that the Drude model with two critical points correction (at least) must be used in general. Double-precision arithmetic is needed to avoid round-off errors if highly converged results are sought. Per-component meshing increases the accuracy when complex geometries are modeled, but the uniform mesh works better for structures completely fillable by the Yee cell (e.g., rectangular structures). Generally, a mesh size of 0.25 nm is required to achieve convergence of results to ∼ 1%. We determine how to optimally setup the simulation domain, and in so doing we find that performing scattering calculations within the near-field does not necessarily produces large errors but reduces the computational resources required.

Subpixel smoothing finite-difference time-domain method for material interface between dielectric and dispersive media

In this Letter, we have shown that the subpixel smoothing technique that eliminates the staircasing error in the finite-difference time-domain method can be extended to material interface between dielectric and dispersive media by local coordinate rotation. First, we show our method is equivalent to the subpixel smoothing method for dielectric interface, then we extend it to a more general case where dispersive/dielectric interface is present. Finally, we provide a numerical example on a scattering problem to demonstrate that we were able to significantly improve the accuracy.

Effective permittivities with exact second-order accuracy at inclined dielectric interface for the two-dimensional finite-difference time-domain method

Accuracy degradation at a dielectric interface in simulations using the finite-difference time-domain method can be prevented by assigning suitable effective permittivities at the nodes in the vicinity of the interface. The effective permittivities with exact second-order accuracy at the interface inclined to the Yee-lattice axis are analytically derived for what we believe to be the first time. We discuss two interfaces with different inclined angles between their normal and the Yee-lattice axis in the case of two-dimensional TE polarization. The tangent of the angle is 1 for one interface and 1/2 for the other. With the derived effective permittivities, reflection and transmission at the interface are simulated with second-order accuracy with respect to cell size. The accuracy is demonstrated by numerical examples.

Spectrum control by anisotropy in a cylindrical microcavity

Spectrum control by anisotropy in a cylindrical microcavity made of electric anisotropic medium was studied. A finite-difference time domain method for electric anisotropic medium and Volume-average Effective Permittivity approximation are applied to calculate the resonant frequencies and quality factors of Whispering-gallery modes. The resonant frequency for different whispering-gallery modes has a similar shift in direct proportion to the relative difference of two principal refractive indices. The quality factors decay exponentially due to directional emission when the difference of two principal refractive indices increases. This novel tuning characteristic of anisotropic cylindrical microcavity will play an important role in many areas, such as light source with tunable wavelength, tunable filter and sensor.

Dispersive contour-path algorithm for the two-dimensional finite-difference time-domain method

We have extended the contour-path effective-permittivity (CP-EP) finite-difference time-domain (FDTD) algorithm by A. Mohammadi, et al., Opt. Express 13, 10367 (2005), to linear dispersive materials using the Z-transform formalism. We test our method against staircasing and the exact solution for plasmon spectra of metal nanoparticles. We show that the dispersive contour-path (DCP) approach yields better results than staircasing, especially for the cancellation of spurious resonances.

Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods

We derive a correct first-order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted. Most previous perturbative methods give incorrect results for this case, even to lowest order, because of the complicated discontinuous boundary conditions on the electric field at such an interface. Our final expression is simply a surface integral, over the material interface, of the continuous field components from the unperturbed structure. The derivation is based on a "localized" coordinate-transformation technique, which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking the limit in which the coordinate perturbation is infinitesimally localized around the boundary. Not only is our result potentially useful in evaluating boundary perturbations, e.g., from fabrication imperfections, in highly anisotropic media such as many metamaterials, but it also has a direct application in numerical electromagnetism. In particular, we show how it leads to a subpixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media, in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes.

Improving accuracy by subpixel smoothing in the finite-difference time domain

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.