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Observation of ultrathin single-domain layers formed on LiTaO3 and LiNbO3 surfaces using scanning nonlinear dielectric microscope with submicron resolution. Scanning electron-beam dielectric microscopy for the investigation of the temperature coefficient distribution of dielectric ceramics. Quantitative measurement of linear and nonlinear dielectric characteristic using scanning nonlinear dielectric microscopy. Higher order nonlinear dielectric microscopy. Eng, L. Ferroelectric domain characterization and manipulation: a challenge for scanning probe microscopy. Nanoscale reconstruction of surface crystallography from three-dimensional polarization distribution in ferroelectric barium-titanate ceramics.

Gao, C. High special resolution quantitative microwave impedance microscopy by a scanning tip microwave near-field microscope. Quantitative microwave near-field microscopy of dielectric properties. Goto, K. Denjikigaku Enshu. Exercise in Electromagnetic Theory , p. Kyoritsu Shuppan, Tokyo. Gruverman, A. Scanning force microscopy of domain structure in ferroelectric thin films: imaging and control.

Nanotechnology 8, A38—A Hatanaka, T. Quasi-phase-mached optical parametric oscillation with periodically poled stoichiometric LiTaO3.

Jona, F. Pergamon Press, New York. Kitamura, K. Matsuura, K. Fundamental study on nano domain engineering using scanning nonlinear dielectric microscopy. Measurement of the ferroelectric domain distributions using nonlinear dielectric response and piezoelectric response. Optical second harmonic generation in piezoelectric crystals. Odagawa, H. Simultaneous observation of nano-sized ferroelectric domains and surface morphology using scanning nonlinear dielectric microscopy.

Surface Science , L—L Theoretical and experimental study on nanoscale ferroelectric domain measurement using scanning nonlinear dielectric microscopy. Measuring ferroelectric polarization component parallel to the surface by scanning nonlinear dielectric microscopy. Ohara, K. Fundamental study of surface layer on ferroelectrics by scanning nonlinear dielectric microscopy. Quantitative measurement of linear dielectric constant using scanning nonlinear dielectric microscopy with electro-conductive cantilever. Pauch, P.

### Lower Division

Nanoscale control of ferroelectric polarization and domain size in epitaxial Pb Zr0. Scattered Field Formulation. Extensions of the Yee Scheme.

Fictitious and Overlapping Grid Methods. Global Methods. Multidomain Formulations. The Local Scheme. Connecting the Elements.

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## Nanophotonics

A Few Examples. Continuous Finite Element Techniques. Discontinuous Finite Element Techniques. As simple as these equations appear, their importance is tremendous and accurate, efficient, and robust methods for solving them are at the heart of the development of emerging technologies, such as very low observable 59 Copyright , Elsevier USA. This imposes requirements on the accuracy and performance of the computational tools well beyond that of existing standard techniques. The need to identify new approaches to electromagnetic modeling and design is further emphasized by the growing interest in very broad band signals and their interaction with large and geometrically complex objects, often including regions of inhomogeneous, anisotropic, lossy, or even nonlinear materials.

As the frequency of the waves increases in applications and in modeling efforts, an additional complication enters through the need to model random surfaces and materials. The classical integral-based solution techniques Chew et al. Finite element techniques Jin, ; Volakis et al. Indeed, the strength of this approach has been demonstrated successfully over the last few decades, beginning with the second-order accurate Yee scheme. As simple as this scheme is, it continues to be the main workhorse of computational electromagnetics in the time domain Kunz and Luebbens, ; Taflove, , It is easy to identify several reasons for the success of the Yee scheme, but its most appealing quality is perhaps its simplicity.

The limitations of the Yee scheme are, however, equally straightforward to identify. Apart from second-order accuracy, limiting the electric size and duration of problems one can consider, the embedding of the computational geometry poses the most significant problem by requiring one to approximate boundaries and interfaces by a staircased curve.

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While this may seem adequate for many problems, it nevertheless affects the overall accuracy and essentially reduces accuracy of the scheme to the first order. Most of these methods, however, sacrifice the simplicity of the original Yee scheme to achieve the improved accuracy, which remains, at best, second order. However, as the problems increase in size and the geometries in complexity, one begins to encounter the limits of the second-order scheme. In particular the accumulating dispersion errors becomes a major concern e.

Ways to overcome these errors are, however, few and well known—decrease the grid size or increase the order of the scheme. As discussed in Section III, high-order methods are characterized by their ability to accurately represent wave propagation over very long distances using only a few points per wavelength. For three-dimensional large-scale computations, this translates into dramatic reductions in the required computational resources, i. This comes at a price, however. The simplicity of the schemes is sacrificed somewhat for the accuracy, particularly when combined with a need for geometric flexibility.

This increased complexity of the scheme is perhaps the main reason for the slow acceptance of high-order methods among practitioners of computational electromagnetics. Although the need for high-order accurate schemes was realized by some practitioners early on Nachman, , acceptance of this is still far from widespread.

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Evidence of this is the lack of contributions discussing high-order time-domain methods in recent overviews of state-of-the-art techniques in computational electromagnetics Graglia et al. By high order we shall refer to methods with a spatial convergence rate exceeding two. The question of which order of accuracy is suitable for large-scale applications is an interesting question in itself and can be analyzed as a cost—benefit question Kreiss and Oliger, ; Fischer and Gottlieb, ; Wasberg and Gottlieb, ; Gottlieb and Hesthaven, While the answer naturally has some problem dependence, the general conclusion is that schemes of spatial order four to six offer an optimal balance between accuracy and computational work for a large class of applications solved with realistic accuracy tolerances.

It is therefore natural to focus on methods that have the potential to reach this level of accuracy. These methods do display high-order accuracy under certain circumstances but are notoriously difficult to use for geometrically complex problems.

## Supercapacitor

Because this remains a major concern in many applications, we have chosen not to include a discussion. A good starting point for such methods is Taflove While more selective overviews are available, we shall strive to bring most current efforts into the discussion. It is hoped that this, one on hand, will be helpful as a starting point to the practitioner seeking alternatives to standard techniques and, on the other hand, can serve as an updated review of an emerging and rapidly evolving field to the interested computational mathematician. This review is organized as follows.

Section III is devoted to an overview of the by now classical phase-error analysis as a way of motivating the need to consider high-order accurate methods in time-domain electromagnetics, particularly as problems increase in size and complexity. This sets the stage for Section IV, which discusses extensions of the Yee scheme and other more complex finite difference schemes.