The CP-FDTD method can still contain cells that potentially generate instability because of non-reciprocal nearest neighbor borrowing steps. For the special cells, a Contour Path (CP) FDTD algorithm can be obtained directly from Maxwell’s equations in integral form. Better is to exploit a Cartesian mesh as much as possible and introduce distorted cells only when it is really necessary. However, while improving the accuracy, such approach not only considerably increases the complexity of the algorithm, but can also cause numerical artifacts due to a highly irregular grid, like time instability, velocity dispersion and spurious wave reflection. This issue has limited the application of the FDTD method when the exact shape and permittivity values are important in determining the electromagnetic response.Ī possible solution to staircasing is to depart from the simple Cartesian mesh picture and use non-orthogonal grids or curvilinear coordinates that follow exactly the shape of the objects. Consequently, the scattering properties of the system will change going from the real shape to the meshed shape, especially if the mesh is coarse and the dielectric contrast is large. Because the mesh can assign only discrete values to position, any object embedded in the simulation domain, will be pixelized in a way that a smooth interface becomes like a staircase. Staircasing can be easily understood by looking at Fig. The two latters will not be directly addressed in this work the reader can refer to a reference book on FDTD. There are, however, some intrinsic problems that can make the FDTD method inaccurate, such as staircasing, numerical velocity dispersion and absorbing boundary conditions. After a few simple rearrangements, the curl equations are finally transformed into loops that simulate the propagation of the electromagnetic field in space and time. The same holds for the time mesh, where the so-called leapfrog scheme is used. The structure of the curl equations suggests that the field components are defined on different discrete positions in space, as shown in Fig. In the FDTD method, space and time are discretized in a way that the derivatives in Maxwell’s curl equations can be written as finite central differences. Yee, has gained much popularity for several reasons: it is rather easy to implement, the algorithm is intuitive, it can solve Maxwell’s equation for systems of arbitrary shape, it works in space and time domain. Among them, the Finite-Difference Time-Domain (FDTD) method, proposed by K.S. In the past, several techniques have been proposed, like finite-difference and finite-element methods, integral equation methods and so forth. These developments rely more and more on complex geometries, like photonic crystals or metamaterials, as well as on complex field configurations, so that full-vector numerical methods play a key role in understanding and designing them.
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Research fields like advanced optical imaging and integrated photonics explore and push forward the terrific possibilities offered by Maxwell’s equations. Significant improvements in terms of accuracy and error fluctuations are demonstrated, especially in the calculation of resonances. Our schemes are validated using Mie theory for the scattering of a dielectric cylinder and they are compared to the usual staircase and the widely used volume-average approximations. A phenomenological formula for the effective permittivities is also proposed as an effective and simpler alternative to the previous result.
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Note the Boolean sign must be in upper-case.