Subpixel Smoothing Research Articles

Using high-resolution microscopy data to generate realistic structures for electromagnetic FDTD simulations from complex biological models.

Finite-difference time-domain (FDTD) electromagnetic simulations are a computational method that has seen much success in the study of biological optics; however, such simulations are often hindered by the difficulty of faithfully replicating complex biological microstructures in the simulation space. Recently, we designed simulations to calculate the trajectory of electromagnetic light waves through realistically reconstructed retinal photoreceptors and found that cone photoreceptor mitochondria play a substantial role in shaping incoming light. In addition to vision research and ophthalmology, such simulations are broadly applicable to studies of the interaction of electromagnetic radiation with biological tissue. Here, we present our method for discretizing complex 3D models of cellular structures for use in FDTD simulations using MEEP, the MIT Electromagnetic Equation Propagation software, including subpixel smoothing at mesh boundaries. Such models can originate from experimental imaging or be constructed by hand. We also include sample code for use in MEEP. Implementation of this algorithm in new code requires understanding of 3D mathematics and may require several weeks of effort, whereas use of our sample code requires knowledge of MEEP and C++ and may take up to a few hours to prepare a model of interest for 3D FDTD simulation. In all cases, access to a facility supercomputer with parallel processing capabilities is recommended. This protocol offers a practical solution to a significant challenge in the field of computational electrodynamics and paves the way for future advancements in the study of light interaction with biological structures.

Implementation of a finite difference frequency domain mode solver incorporating subpixel smoothing.

Finite difference frequency domain (FDFD) mode solvers are straightforward to implement but can suffer from slow convergence when applied to high-contrast refractive index structures. In this work, we show how subpixel smoothing can improve the convergence properties of a full-vectorial FDFD mode solver. Based on a standard Yee grid, we formulate a generalized eigenbproblem whose solutions provide the modes of the waveguides taking into account the tensor nature of the effective dielectric constant. We investigate the convergence of the proposed FDFD mode solver in several cases including a step index fiber, a microsctuctured fiber, and a cylindrical plasmonic waveguide. The results show that tensor smoothing can significantly improve the convergence of the solver, thus allowing the use of less dense grids in the calculations. Our implementation is freely available on the web under an open-source licence.

Improving accuracy using subpixel smoothing for multiband effective-mass Hamiltonians of semiconductor nanostructures

We develop schemes of subpixel smoothing for the multiband Luttinger–Kohn and Burt–Foreman Hamiltonians of semiconductor nanostructures. With proper procedures of parameter averages at abrupt interfaces, computational errors of envelope functions due to the discontinuity of heterostructures are significantly reduced. Two smoothing approaches are presented. One is based on eliminations of the first-order perturbation in energy, and the other is an application of the Hellmann–Feynman theorem. Using the finite-difference method, we find that while the procedure of perturbation theory seems to be more robust than that of Hellmann–Feynman theorem, the errors of both schemes are (considerably) lower than that without smoothing or with direct but unjustified averages of untransformed parameters. The proposed approaches may enhance numerical accuracies and reduce computational cost for the modeling of nanostructures.

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.

Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing

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.

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.

Subpixel smoothing for conductive and dispersive media in the finite-difference time-domain method

Staircasing of media properties is one of the intrinsic problems of the finite-difference time-domain method, which reduces its accuracy. There are different approaches for solving this problem, and the most successful of them are based on correct approximation of inverse permittivity tensor epsilon(-1) at the material interface. We report an application of this tensor method for conductive and dispersive media. For validation, comparisons with analytical solutions and various other subpixel smoothing methods are performed for the Mie scattering from a small sphere.

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.