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The current rapid and complex advancement applications of electromagnetic (EM ) and optical systems calls for a much needed update on the computational.

**Table of contents**

- Computational Methods for Electromagnetic and Optical Systems
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- Computational Methods for Electromagnetic and Optical Systems

It can be applied in many areas of engineering and science including fluid mechanics , acoustics , electromagnetics , fracture mechanics , and plasticity. MoM has become more popular since the s.

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Conceptually, it works by constructing a "mesh" over the modeled surface. However, for many problems, BEM are significantly computationally less efficient than volume-discretization methods finite element method , finite difference method , finite volume method. Boundary element formulations typically give rise to fully populated matrices.

This means that the storage requirements and computational time will tend to grow according to the square of the problem size. By contrast, finite element matrices are typically banded elements are only locally connected and the storage requirements for the system matrices typically grow linearly with the problem size.

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Compression techniques e. BEM is applicable to problems for which Green's functions can be calculated. These usually involve fields in linear homogeneous media. This places considerable restrictions on the range and generality of problems suitable for boundary elements. Nonlinearities can be included in the formulation, although they generally introduce volume integrals which require the volume to be discretized before solution, removing an oft-cited advantage of BEM.

It is an accurate simulation technique and requires less memory and processor power than MoM. The FMM was first introduced by Greengard and Rokhlin [3] [4] and is based on the multipole expansion technique. The first application of the FMM in computational electromagnetics was by Engheta et al. While the fast multipole method is useful for accelerating MoM solutions of integral equations with static or frequency-domain oscillatory kernels, the plane wave time-domain PWTD algorithm employs similar ideas to accelerate the MoM solution of time-domain integral equations involving the retarded potential.

The partial element equivalent circuit PEEC is a 3D full-wave modeling method suitable for combined electromagnetic and circuit analysis. The equivalent circuit formulation allows for additional SPICE type circuit elements to be easily included. Further, the models and the analysis apply to both the time and the frequency domains.

Besides providing a direct current solution, it has several other advantages over a MoM analysis for this class of problems since any type of circuit element can be included in a straightforward way with appropriate matrix stamps. The PEEC method has recently been extended to include nonorthogonal geometries. This helps in keeping the number of unknowns at a minimum and thus reduces computational time for nonorthogonal geometries.

It is easy to understand. It has an exceptionally simple implementation for a full wave solver. FDTD is the only technique where one person can realistically implement oneself in a reasonable time frame, but even then, this will be for a quite specific problem. FDTD belongs in the general class of grid-based differential time-domain numerical modeling methods. Maxwell's equations in partial differential form are modified to central-difference equations, discretized, and implemented in software. The equations are solved in a cyclic manner: the electric field is solved at a given instant in time, then the magnetic field is solved at the next instant in time, and the process is repeated over and over again.

Since about , FDTD techniques have emerged as the primary means to model many scientific and engineering problems addressing electromagnetic wave interactions with material structures. An effective technique based on a time-domain finite-volume discretization procedure was introduced by Mohammadian et al. Approximately 30 commercial and university-developed software suites are available. Like FVTD, the numerical flux is used to exchange information between neighboring elements, thus all operations of DGTD are local and easily parallelizable.

With above merits, DGTD method is widely implemented for the transient analysis of multiscale problems involving large number of unknowns. The finite element method FEM is used to find approximate solution of partial differential equations PDE and integral equations. The solution approach is based either on eliminating the time derivatives completely steady state problems , or rendering the PDE into an equivalent ordinary differential equation , which is then solved using standard techniques such as finite differences , etc.

In solving partial differential equations , the primary challenge is to create an equation which approximates the equation to be studied, but which is numerically stable , meaning that errors in the input data and intermediate calculations do not accumulate and destroy the meaning of the resulting output.

There are many ways of doing this, with various advantages and disadvantages. The finite element method is a good choice for solving partial differential equations over complex domains or when the desired precision varies over the entire domain. The finite integration technique FIT is a spatial discretization scheme to numerically solve electromagnetic field problems in time and frequency domain.

It preserves basic topological properties of the continuous equations such as conservation of charge and energy.

## Computational Methods for Electromagnetic and Optical Systems

FIT was proposed in by Thomas Weiland and has been enhanced continually over the years. The basic idea of this approach is to apply the Maxwell equations in integral form to a set of staggered grids. This method stands out due to high flexibility in geometric modeling and boundary handling as well as incorporation of arbitrary material distributions and material properties such as anisotropy , non-linearity and dispersion. Furthermore, the use of a consistent dual orthogonal grid e. Cartesian grid in conjunction with an explicit time integration scheme e.

This class of marching-in-time computational techniques for Maxwell's equations uses either discrete Fourier or discrete Chebyshev transforms to calculate the spatial derivatives of the electric and magnetic field vector components that are arranged in either a 2-D grid or 3-D lattice of unit cells. PSTD causes negligible numerical phase velocity anisotropy errors relative to FDTD, and therefore allows problems of much greater electrical size to be modeled. PSSD solves Maxwell's equations by propagating them forward in a chosen spatial direction.

The fields are therefore held as a function of time, and possibly any transverse spatial dimensions. The method is pseudo-spectral because temporal derivatives are calculated in the frequency domain with the aid of FFTs.

Because the fields are held as functions of time, this enables arbitrary dispersion in the propagation medium to be rapidly and accurately modelled with minimal effort. This is an implicit method. In this method, in two-dimensional case, Maxwell equations are computed in two steps, whereas in three-dimensional case Maxwell equations are divided into three spatial coordinate directions. Eigenmode expansion EME is a rigorous bi-directional technique to simulate electromagnetic propagation which relies on the decomposition of the electromagnetic fields into a basis set of local eigenmodes.

The eigenmodes are found by solving Maxwell's equations in each local cross-section. Eigenmode expansion can solve Maxwell's equations in 2D and 3D and can provide a fully vectorial solution provided that the mode solvers are vectorial. It offers very strong benefits compared with the FDTD method for the modelling of optical waveguides, and it is a popular tool for the modelling of fiber optics and silicon photonics devices. Physical optics PO is the name of a high frequency approximation short- wavelength approximation commonly used in optics, electrical engineering and applied physics.

It is an intermediate method between geometric optics, which ignores wave effects, and full wave electromagnetism , which is a precise theory. The word "physical" means that it is more physical than geometrical optics and not that it is an exact physical theory. The approximation consists of using ray optics to estimate the field on a surface and then integrating that field over the surface to calculate the transmitted or scattered field. This resembles the Born approximation , in that the details of the problem are treated as a perturbation.

The uniform theory of diffraction UTD is a high frequency method for solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in more than one dimension at the same point. The uniform theory of diffraction approximates near field electromagnetic fields as quasi optical and uses ray diffraction to determine diffraction coefficients for each diffracting object-source combination.

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These coefficients are then used to calculate the field strength and phase for each direction away from the diffracting point. These fields are then added to the incident fields and reflected fields to obtain a total solution.

Validation is one of the key issues facing electromagnetic simulation users. The user must understand and master the validity domain of its simulation. The measure is, "how far from the reality are the results? Answering this question involves three steps: comparison between simulation results and analytical formulation, cross-comparison between codes, and comparison of simulation results with measurement.

## Computational Methods for Electromagnetic and Optical Systems

For example, assessing the value of the radar cross section of a plate with the analytical formula:. Computational complexity of solving integral equations. Sparsification of matrices by designing basis and testing functions: impedance matrix localization and complex multipole beam approach.

Fast field evaluation algorithms: FFT, fast multipole method and non-uniform grid approach. Surface integral equations for three-dimensional problems. Discretization using RWG basis functions. Textbooks Peterson, A. Ray, and R. Computational Methods for Electromagnetics. IEEE Press, Harrington, R. Field Computation by Moment Methods. Mc-Millan, Chew W. Jin, E. Michielssen, and J.

Fast and Efficient Algorithms in Computational Electromagnetics. Artech House Selected journal papers. Homework assignments will be given during the week.