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General Principles of the Wave Concept Iterative Process

1.1. Introduction

The iterative method, which uses a wave network, is an integrated method and is not based upon electric and magnetic fields, as are, for example, Electrical Field Integral Equation (EFIE), Magnetic Field Integral Equation (MFIE), or more generally the method of moments or a combination of both fields. These are likened to the amplitudes of transverse waves, both diffracting around obstacles and those in space, termed "free space", owing to the presence of evanescent fields. However, while the method of moments appeals to so-called admittance or impedance operators, within the wave iterative method (Wave Concept Iterative Process (WCIP)), the diffraction operators are restricted, thus leading to the convergence of all iterative processes based upon this particular formalism [BAU 99].

It may be noted that, with the method of moments, the solution to the problem often entails using a restriction in the given field so as to define trial functions that constitute the basis for given solutions. This often leads to both analytical and numerical problems. In the WCIP method, field conditions are simply described on the basis of pixels which make up the entire sphere.

Moreover, the iterative process has a significant resemblance to that used within harmonic equilibrium [KER 75]. Within this latter process the nonlinear component behaves in a way that is described in relation to time, while the rest of the circuit is described within the frequency sphere. The operator thus functions diagonally at given frequencies. With each iteration, we therefore proceed with a Fourier transform (using a time-frequency basis) so as to approach the detailed composition of boundary conditions at the shutdown level. Moreover, when writing equations in terms of components studied over time, an inverse Fourier transform (based upon frequency-time) is used.

The WCIP approach is closely related. By simply replacing time by a coordinate and the frequency by a "spatial frequency", the operation reverts to one within the spectral sphere. Outside of the Transverse Lines Matrix (TLM) method, which also necessitates the wave concept [KRU 94], the WCIP is based upon the systematic iteration between both incident and reflected waves. The approach used in the paragraphs below is as follows: select a wave definition which is consistent with pre-existing cases, in particular within waveguides, and ensure that it has a fundamental physical significance. The iterative process will then be described in the context of several types of problems, in particular quasi-periodic structures.

The objectives of this chapter are to first set out the WCIP, showing its potential for circuit modeling, antennae and quasi-optical devices within stratified environments [BAU 99, AZI 95, AZI 96, WAN 05, RAV 04, TIT 09]. There are two advantages to this method. Firstly, the iterative process is always convergent (excepting the frequency resonating from a mechanism such as that one which is also relevant to other digital methods). Secondly, by the description of all surfaces through the use of pixels, it is not necessary to use a network describing the part of the surface corresponding to a metallic coating (or indeed to the dielectric dual), as falls within the sphere of the method of moments.

In the second part of this chapter, the WCIP is outlined. The principles of the WCIP are adhered to. Through the use of combined equations, one is expressed in the spatial sphere and the other in the spectral sphere (also called the modal sphere). The solution is obtained by achieving equilibrium between these two spheres. The description of a given mechanism is not set by rectangular pixels but by cells restricted by periodic barriers, each containing periodic non-configured sources. The sources are described within the spectral sphere (defined by periodic barriers). They are called "auxiliary sources", as they need to be substituted by impedances alone or indeed other sources or impedances, the latter being defined by each cell. Hence the use of the term spatial domain and the adjective quasi-periodic being applied to the system. This concept makes it possible to study a large variety of systems, in particular Substrate Integrated Circuits (SIC), which have been successfully developed for several years, as shown by the results from a number of examples.

The last part of the chapter provides a gateway to other interesting applications in the field of non-homogeneous meta-materials, in the sense that both sources and obstacles are integrated in a unique model, thus avoiding the use of the approximation of equivalent environments. Up until now (except for using three-dimensional simulators such as EF and Finite-Difference Time-Domain (FDTD)) homogeneous meta-materials allow us to establish their equivalent index. The link between a material and a given mechanism (for example, plate antennae) presents difficulties and cannot be approached from any perspective other than a comprehensive analysis.

Finally, we provide an overview of another WCIP-based field; the study of quasi-periodic circuits with identified components. There are many applications for these types of structures, filters, amplifiers, percolation problems and quasi-optic planar sources.

1.2. The iterative wave method

The integral form of waves came to be explained during the 1990s, and was applied to planar circuits and to antennae [BAU 99, AZI 95, AZI 96, WAN 05, RAV 04, TIT 09]. The wave concept principle is as follows:

1. - The electromagnetic issue may be expressed by the relationship between the two environments. The first is known as the spectral sphere or the external environment. The second is a set of surfaces which are defined by the boundary conditions at each point (termed the spatial domain). An Ao source in the spatial sphere sends a wave with an Ao amplitude towards a vacuum of free space. This wave is partly reflected (by the reaction of the operator G) and provides a wave B. The latter is, in its turn, reflected within the spatial sphere (the Operator S) giving us the wave A.
2. - The G operator is diagonal within the spectral sphere. It represents the homogeneous environment and its interaction with electromagnetic waves [BAU 99]. The operator S describes the boundary conditions of the interface. It is expressed within the spatial sphere. The Fourier transform and its converse, the inverse Fourier transform, ensure the passage between both spheres. The relationships between incident and reflective waves are written as shown in [1.1] and [1.2].

With the first iteration, the spatial sphere equation should be expressed simply as Ao (B = 0). B now appears with the operator G (B = GA). The equation [1.2] is applied so as to obtain the new value of A placed within [1.1], resulting in the new B value. This iterative process consists in successively applying equations [1.1] and [1.2], until convergence occurs (Figure 1.1).

Figure 1.1. Iterative wave diagram

For a planar circuit within a rectangular housing, the operator G is diagonal across modes TE and TM, as are both the rapid transformation in methods Fast mode transform (FMT) and the opposing linking equations [1.1] and [1.2], this occurs because the operator S is diagonal in the spatial domain.

1.3. General definition of waves

The general definition of waves must meet certain conditions:

The existence of a division of the overall sphere into two sub-spheres: the internal sphere or the spatial sphere (these are flat interfaces or localized elements, indeed centers of boundary conditions within integrated methods). The second sphere is the external sphere (or spectral sphere). This sphere is most often described on the basis of the unique functions of the Helmholtz operator, which stems from Maxwell's equations. To develop this method, we need to define two dual variables such as Current-Voltage, Electric field-Magnetic field, Current density (density or surface)-Electric field, and Voltage-Load density or Voltage-Load. All of the possibilities are shown in Table 1.1. E and J may be taken as two dual variables. J is not necessarily a current-related density, but encompasses all magnitudes which are defined in Table 1.1. J may also be related to current volume density. One would thus write it as Jv to avoid confusion with the magnetic field rotated by 90°(H^n). Wave amplitudes A and B are thus defined (it may be observed that A and B may be scalars or vectors):

1.4. Application to planar circuits

The first and most frequent wave representation is where dual variables are tangential components of fields on a surface S (Figure 1.2) adjusted to account for the magnetic field by H^n (the vector product is chosen rather than the magnetic field, simply for reasons of homogeneity). This representation is highly useful when dealing with circuits and planar antennae, frequency-selective surfaces, diffraction issues and systems involving cylindrical or spherical coordinates.

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