Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section and for Complementary Shape of the Cross Section

Abstract

This chapter presents wave propagation along a straight rectangular waveguide for practical applications where there are two complementary shapes of the dielectric profile in the cross section. In the first case, the cross section consists of circular dielectric material in the center of the cross section. In the second case, the cross section consists of a circular hollow core in the center of the cross section. These examples show two discontinuous cross sections and complementary shapes that cannot be solved by analytical methods. We will explain in detail the special technique for calculating the dielectric profile for all cases. The method is based on Laplace and Fourier transforms and inverse Laplace and Fourier transform. In order to solve any inhomogeneous problem in the cross section, more than one technique can be proposed for the same mode-model method. We will explain in detail how and where the technique can be integrated into the proposed mode-model. The image method and periodic replication are needed for fulfilling the boundary condition of the metallic waveguide. The applications are useful for straight rectangular waveguides in millimeter regimes, where the circular dielectric material is located in the center of the cross section, and also for hollow waveguides, where the circular hollow core is located in the center of the cross section.

1. Introduction

We begin with a review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides with emphasis on recent developments as published in [1]. Six groups of methods were reviewed: the finite-element method, the finite-difference method, the integral-equation method, methods based on series expansion, approximate methods based on separation of variables, and methods that do not fit the above groups.

The use of wavelet-like basis functions for solving electromagnetics problems is demonstrated in [2]. The modes of an arbitrarily shaped hollow metallic waveguide use a surface integral equation and the method of moments. A class of wavelet-like basis functions produces a sparse method of moments. A technique for efficient computation of an integral wavelet transform of a finite-energy function on a dense set of the time-scale domain is proposed [3] by using compactly supported spline wavelets. Application of principal component analysis and wavelet transform to fatigue crack detection in waveguides is proposed in [4]. Ultrasonic guided waves are a useful tool in structural health monitoring applications that can benefit from built-in transduction, moderately large inspection ranges, and high sensitivity to small flaws. An accurate full-wave integral formulation was developed [5] for the study of integrated planar dielectric waveguide structures with printed metalized sections, which are of practical interest for millimeter-wave and submillimeter-wave applications. An advantageous finite element method for the rectangular waveguide problem was developed [6] by which complex propagation characteristics may be obtained for arbitrarily shaped waveguides. The finite-element method has been used to derive approximate values of the possible propagation constant for each frequency. The impedance characteristics of the fundamental mode in a rectangular waveguide were computed using this finite element method. The extension to higher-order elements is straightforward, and by modifications of the method it is possible to treat other types of waveguides as well, e.g., dielectric waveguides with impedance walls and open unbounded dielectric waveguides properties treating the region of infinity.

A comprehensive study of the design and performance of a multilayer dielectric rod waveguide with a rectangular cross section is proposed in [7]. The design is comprised of a high permittivity core encased by a low permittivity cladding. A mathematical model was proposed to predict the fundamental mode cutoff frequency in terms of the core dimensions and the core and cladding permittivity. The model is useful for design purposes and it offers an excellent match to full-wave electromagnetic simulation results.

The characteristics of the effective-medium-clad dielectric waveguides, including dispersion, cross-polarization, crosstalk between parallel waveguides, bending loss, and wave leakage at the crossing, have been comprehensively investigated and measured [8].

Mode matching has been done at all the air and dielectric interfaces and thus the characteristic equations have been derived [9]. Two ratios are introduced in the characteristic equations and the new set of characteristic equations thus obtained are then plotted and graphical solutions are obtained for the propagation parameters assuming certain numerical values for the introduced ratios.

A fundamental and accurate technique to compute the propagation constant of waves in a dielectric rectangular waveguide was proposed [10]. The formulation is based on matching the fields to the constitutive properties of the material at the boundary.

The method of lines for the analysis of dielectric waveguides was proposed [11]. These waveguides are uniform along the direction of propagation, are loss-free and passive. Hybrid-mode dispersion curves, field and intensity distributions for integrated optical waveguides were presented.

The problem of normal waves in a closed regular waveguide of arbitrary cross section has been considered [12]. It was reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The solutions were defined using the variational formulation of the problem. The problem was reduced to the study of an operator function. The properties of the operators involved in the operator function were examined. Theorems were proved concerning the discrete character of the spectrum and the distribution of characteristic numbers of the operator function on the complex plane. The completeness of the system of Eigen- and associated vectors of the operator function was investigated.

TE-wave propagation in a hollow waveguide with a graded dielectric layer using a hyperbolic tangent function is proposed in Ref. [13]. General formulas for the electric field components of the TE-waves, applicable to hollow waveguides with arbitrary cross sectional shapes were presented. The exact analytical results for the electric field components were illustrated in the special case of a rectangular waveguide. The exact analytical results for the reflection and transmission coefficients are valid for waveguides of arbitrary cross sectional shapes. The obtained reflection and transmission coefficients are in exact asymptotic agreement with those obtained for a very thin homogeneous dielectric layer using mode-matching and cascading. The proposed method gives analytical results that are directly applicable without the need of mode-matching, and it has the ability to model realistic, smooth transitions.

Rectangular waveguides were the earliest mode of transmission lines used for compact systems like radars and inside equipment shelters [14]. An air-filled rectangular waveguide WR-90 is simulated using HFSS simulation software to obtain different parameters. The electric and magnetic field patterns are analyzed: intrinsic impedance and wavelength for the first four modes of the waveguide are also obtained.

The diffraction of electromagnetic waves by rectangular waveguides with a longitudinal slit has been simulated [15]. The results allow determining the patterns of change in frequency bands in which the structure can be used as a directional coupler and as a power divider when changing the number of slots, their sizes and provisions. Modeling the characteristics of such kinds of structures allows predicting the creation of directional couplers and power dividers with high integral characteristics.

Several methods of propagation along the straight waveguides were developed, based on Maxwell’s equations. A transfer matrix function for the analysis of electromagnetic wave propagation along the straight dielectric waveguide with arbitrary profiles has been proposed in Ref. [16].

In this chapter, the main objective is to generalize the mode model method [16] in order to solve also complicated and practical problems of circular dielectric material and a circular hollow core in the center of the cross section of the rectangular waveguide. It is important to distinguish between the mode-model method and the proposed technique. The proposed technique deals only with calculating the dielectric profile in the cross section of the inhomogeneous case. In order to solve any inhomogeneous problem in the cross section, more than one technique can be proposed for the same mode-model method. The technique proposed in this chapter will refer to two interesting practical applications. In the first case, the cross section consists of circular dielectric material in the center of the cross section. In the second case, the cross section shows the complementary shape of the cross section of the first case, as an example of a hollow waveguide in which the circular hollow core is located in the center of the cross section. These examples show two discontinuous cross sections and complementary shapes that cannot be solved by analytical methods. We will explain in detail the special technique for calculating the dielectric profile in all cases. After receiving the expressions of the proposed technique for each inhomogeneous problem in the cross section, we will explain how and where the technique can be integrated into the proposed mode-model. The second objective is to find the relevant parameters in order to obtain the Gaussian behavior of the output field in the interesting cases of circular dielectic material and a circular hollow core in the rectangular cross section.

2. Complementary shapes in the cross section for different applications

The wavelet transform creates a representation of the signal in both the time and frequency domain in order to allow efficient access to localized information about the signal. A set of waveforms comprising a transform is called a basis function. Fourier transforms use only sine and cosine waves as their basis functions, namely a signal is decomposed into a series sine and cosine functions or wavelets by the FFT. Examples for the applications of wavelet transform are demonstrated in [2, 3, 4, 5]. The proposed method in this chapter is based on the Fourier transform that creates a representation of the signal in the frequency domain. Two complicated and complementary shapes are given in this section.

Figure 1(a) and (b) shows two complementary shapes of profiles in the cross section of the straight rectangular waveguide and their relevant parameters. The circular dielectric material in the center of the cross section is shown in Figure 1(a) and the circular hollow core in the center of the cross section is shown in Figure 1(b). The two examples are demonstrated as a response to a half-sine ($\mathrm{T}{\mathrm{E}}_{10}$) input-wave profile. These two different complementary shapes of the cross section are demonstrated for two different applications. The first example (Figure 1(a)) is useful in millimeter regimes where the circular dielectric material is located in the center of the cross section. The second example (Figure 1(b)) is useful in the millimeter regimes where the circular hollow core is located in the center of the cross section.

Figure 1.

Complementary shapes of profiles in the cross section of the straight rectangular waveguide and their relevant parameters. (a) Circular dielectric material in the center of the cross section. (b) A circular hollow core in the center of the cross section.

The main objective is to generalize the mode model method [16] in order to also solve complicated problems of circular dielectric material and a circular hollow core in the rectangular cross section. All the mathematical development relates to the frequency domain. The main points are given in Appendix A.

It is important to separate between mode-model method and the proposed technique. The proposed technique refers only to calculating of the dielectric profile in the cross section of the inhomogeneous problem. In order to solve any inhomogeneous problem in the cross section, more than one technique can be proposed for the same mode-model method. After receiving the expressions of the proposed technique for eace inhomogeneous problem in the cross section, we will explain how and where the technique can be integrated into the proposed mode-model. The second objective is to find the relevant parameters in order to obtain the Gaussian behavior of the output field in the interesting cases of circular dielectic profile and circular hollow profile in the rectangular cross section.

The method is based on Maxwell’s equations for the computation of output fields at each point along the straight waveguide. This method relates the wave profile at the output to the input wave in the Laplace space. A Laplace transform is necessary to obtain convenient and simple input–output connections of the fields. The method consists of Fourier coefficients of the transverse dielectric profile and of the input–output profile. Thus, the accuracy of the method depends on the number of the modes in the system.

The output transverse field profiles are computed by the inverse Laplace and Fourier transforms. The output components of the electric field are given finally by

$${\mathbf{E}}_{\mathbf{x}}={\left\{{\mathbf{D}}_{\mathbf{x}}+{\alpha }_{\mathbf{1}}{\mathbf{M}}_{\mathbf{1}}{\mathbf{M}}_{\mathbf{2}}\right\}}^{-1}\left({\widehat{\mathbf{E}}}_{{\mathbf{x}}_{\mathbf{0}}}-{\alpha }_2{\mathbf{M}}_1{\widehat{\mathbf{E}}}_{{\mathbf{y}}_{\mathbf{0}}}\right),$$E1

$${\mathbf{E}}_{\mathbf{y}}={\left\{{\mathbf{D}}_{\mathbf{y}}+{\alpha }_{\mathbf{1}}{\mathbf{M}}_{\mathbf{3}}{\mathbf{M}}_{\mathbf{4}}\right\}}^{-1}\left({\widehat{\mathbf{E}}}_{{\mathbf{y}}_{\mathbf{0}}}-{\alpha }_{\mathbf{3}}{\mathbf{M}}_{\mathbf{3}}{\widehat{\mathbf{E}}}_{{\mathbf{x}}_{\mathbf{0}}}\right),$$E2

$${\mathbf{E}}_{\mathbf{z}}={\mathbf{D}}_{\mathbf{z}}^{-\mathbf{1}}\left\{{\widehat{\mathbf{E}}}_{{\mathbf{z}}_{\mathbf{0}}}+\frac{\mathbf{1}}{\mathbf{2}\mathbf{s}}\left({\mathbf{G}}_{\mathbf{x}}{\mathbf{E}}_{{\mathbf{x}}_{\mathbf{0}}}+{\mathbf{G}}_{\mathbf{y}}{\mathbf{E}}_{{\mathbf{y}}_{\mathbf{0}}}\right)-\frac{1}{2}\left({\mathbf{G}}_{\mathbf{x}}{\mathbf{E}}_{\mathbf{x}}+{\mathbf{G}}_y{\mathbf{E}}_{\mathbf{y}}\right)\right\},$$E3

where ${\mathbf{E}}_{{\mathbf{x}}_{\mathbf{0}}},{\mathbf{E}}_{{\mathbf{y}}_{\mathbf{0}}},{\mathbf{E}}_{{\mathbf{z}}_{\mathbf{0}}}$ are the initial values of the corresponding fields at z = 0, i.e., ${\mathbf{E}}_{{\mathbf{x}}_{\mathbf{0}}}={\mathbf{E}}_{\mathbf{x}}$ (x, y, z = 0), and ${\widehat{\mathbf{E}}}_{{\mathbf{x}}_{\mathbf{0}}},{\widehat{\mathbf{E}}}_{{\mathbf{y}}_{\mathbf{0}}},{\widehat{\mathbf{E}}}_{{\mathbf{z}}_{\mathbf{0}}}$ are the initial-value vectors.

The modified wave-number matrices are given by

$$\begin{array}{l}{\mathbf{D}}_{\mathbf{x}}\equiv {\mathbf{K}}^{\left(0\right)}+\frac{k_o^2{\chi }_0}{2s}\mathbf{G}+\frac{{\mathit{jk}}_{\mathit{ox}}}{2s}{\mathbf{NG}}_{\mathbf{x}},{\mathbf{D}}_{\mathbf{y}}\equiv {\mathbf{K}}^{\left(0\right)}+\frac{k_o^2{\chi }_0}{2s}\mathbf{G}+\frac{{\mathit{jk}}_{\mathit{oy}}}{2s}{\mathbf{MG}}_{\mathbf{y}},\\ {\mathbf{D}}_{\mathbf{z}}\equiv {\mathbf{K}}^{\left(0\right)}+\frac{k_o^2{\chi }_0}{2s}\mathbf{G},\end{array}$$E4

where the diagonal matrices ${\mathbf{K}}^{\left(\mathbf{0}\right)}$, $\mathbf{M}$, and $\mathbf{N}$ are given by

$$\begin{array}{c}{\mathbf{K}}_{\left(\mathbf{n},\mathbf{m}\right)\left({\mathbf{n}}^{\prime },{\mathbf{m}}^{\prime }\right)}^{\left(\mathbf{0}\right)}=\left\{\left[{\mathbf{k}}_{\mathbf{o}}^{\mathbf{2}}-{\left(\mathbf{n}\pi /\mathbf{a}\right)}^{\mathbf{2}}-{\left(\mathbf{m}\pi /\mathbf{b}\right)}^{\mathbf{2}}+{\mathbf{s}}^{\mathbf{2}}\right]/\mathbf{2}\mathbf{s}\right\}{\delta }_{\mathbf{n}{\mathbf{n}}^{\prime }}{\delta }_{\mathbf{m}{\mathbf{m}}^{\prime }},\\ {\mathbf{M}}_{\left(\mathbf{n},\mathbf{m}\right)\left({\mathbf{n}}^{\prime },{\mathbf{m}}^{\prime }\right)}=\mathbf{m}{\delta }_{\mathbf{n}{\mathbf{n}}^{\prime }}{\delta }_{\mathbf{m}{\mathbf{m}}^{\prime }},{\mathbf{N}}_{\left(\mathbf{n},\mathbf{m}\right)\left({\mathbf{n}}^{\prime },{\mathbf{m}}^{\prime }\right)}=\mathbf{n}{\delta }_{\mathbf{n}{\mathbf{n}}^{\prime }}{\delta }_{\mathbf{m}{\mathbf{m}}^{\prime }},\end{array}$$E5

and where

$$\begin{array}{c}{\alpha }_1=\frac{k_{\mathit{ox}}{k}_{\mathit{oy}}}{4s^2},{\alpha }_2=\frac{{\mathit{jk}}_{\mathit{ox}}}{2s},{\alpha }_3=\frac{{\mathit{jk}}_{\mathit{oy}}}{2s},\\ {\mathbf{M}}_{\mathbf{1}}={\mathbf{NG}}_{\mathbf{y}}{\mathbf{D}}_{\mathbf{y}}^{-1},{\mathbf{M}}_{\mathbf{2}}={\mathbf{MG}}_{\mathbf{x}},{\mathbf{M}}_{\mathbf{3}}={\mathbf{MG}}_{\mathbf{x}}{\mathbf{D}}_{\mathbf{x}}^{-1},{\mathbf{M}}_{\mathbf{4}}={\mathbf{NG}}_{\mathbf{y}}.\end{array}$$E6

Similarly, the other components of the magnetic field are obtained. The output transverse field profiles are given by the inverse Laplace and Fourier transforms, as follows

$${\mathbf{E}}_{\mathbf{y}}\left(\mathbf{x},\mathbf{y},\mathbf{z}\right)=\sum _{\mathbf{n}}\sum _{\mathbf{m}}{\int }_{\sigma -{\mathbf{j}}_{\infty }}^{\sigma +{\mathbf{j}}_{\infty }}{\mathbf{E}}_{\mathbf{y}}\left(\mathbf{n},\mathbf{m},\mathbf{s}\right)exp\left[{\mathbf{jnk}}_{\mathbf{ox}}\mathbf{x}+{\mathbf{jmk}}_{\mathbf{oy}}\mathbf{y}+\mathbf{sz}\right]\mathbf{ds},$$E7

where the inverse Laplace transform is calculated according to the Salzer method [17, 18]. The inverse Laplace transform is performed in this study by a direct numerical integration on the Laplace transform domain by using the method of Gaussian Quadrature. The integration path in the right side of the Laplace transform domain includes all the singularities.

$${\int }_{\sigma -j_{\infty }}^{\sigma +j_{\infty }}{e}^{\mathit{s\zeta }}{E}_y\left(s\right)\mathit{ds}=\frac{1}{\zeta }{\int }_{\sigma -j_{\infty }}^{\sigma +j_{\infty }}{e}^p{E}_y\left(p/\zeta \right)\mathit{dp}=\frac{1}{\zeta }\sum _{i=1}^{15}{w}_i{E}_y\left(s=p_i/\zeta \right),$$E8

where $w_i$ and $p_i$ are the weights and zeros, respectively, of the orthogonal polynomials of order 15. The Laplace variable $s$ is normalized by $p_i/\zeta$ in the integration points, where Re$\left(p_i\right)>0$ and all the poles should be localized on their left side on the Laplace transform domain. This approach of a direct integral transform does not require as in other methods, to deal with each singularity separately.

The relation between the functions f (t) and F (p) is given by

$$f\left(t\right)=\frac{1}{2\mathit{\pi j}}{\int }_{\sigma -j_{\infty }}^{\sigma +j_{\infty }}{e}^{\mathit{pt}}F\left(p\right)\mathit{dp}.$$E9

The function F(p) may be either known only numerically or too complicated for evaluating f(t) by Cauchy’s theorem. The function F(p) behaves like a Polynomial without a constant term, in the variable 1/p, along ($\sigma -j_{\infty }$, $\sigma +j_{\infty }$). One may find f(t) numerically by using new quadrature formulas (analogous to those employing the zeros of the Laguerre polynomials in the direct Laplace transform). A suitable choice of $p_i$ yields an n-point quadrature formula that is exact when $p_{2}n$ is any arbitrary polynomial of the 2n(th) degree in $x\equiv 1/p$, namely

$$\frac{1}{2\mathit{\pi j}}{\int }_{\sigma -j_{\infty }}^{\sigma +j_{\infty }}{e}^p\rho \left(1/p\right)\mathit{dp}=\sum _{i=1}^n{A}_i\left(n\right){\rho }_{2n}\left(1/p_i\right).$$E10

In Eq. (10), $x_i\equiv 1/p_i$ are the zeros of the orthogonal polynomials $p_n\left(x\right)\equiv \mathrm{\Pi }\left(x-x_i\right)$ where

$$\frac{1}{2\mathit{\pi j}}{\int }_{\sigma -j_{\infty }}^{\sigma +j_{\infty }}{e}^p\left(\frac{1}{p}\right)p_n\left(\frac{1}{p}\right){\left(\frac{1}{p}\right)}^i\mathit{dp}=0,$$E11

i = 0,1,…,n − 1 and $A_i\left(n\right)$ correspond to the Christoffel numbers. The normalization $P_n\left(1/p\right)\equiv \left(4n-2\right)\left(4n-6\right),\dots ,6p_n\left(1/p\right)$, for $n\ge 2$, produces all integral coefficients. $P_n\left(1/p\right)$ is proven to be ${\left(-1\right)}^n{e}^{-p}{p}^n{d}^n\left(e^p/p^n\right)/{\mathit{dp}}^n$. The numerical table gives us the values of the reciprocals of the zeros of $P_n\left(x\right)$ or $p_i\left(n\right)$, the zeros of $P_n\left(x\right)$, or $1/{p_i}^n$, and the corresponding Christoffel numbers $A_i\left(n\right)$. By using these quantities in the quadrature formula that represents in Eq. (10), then the “Christoffel numbers” are given by

$$A_i\left(n\right)\equiv \frac{1}{2\mathit{\pi j}}{\int }_{\sigma -j_{\infty }}^{\sigma +j_{\infty }}{e}^p{L}_i\left(n+1\right)\left(\frac{1}{p}\right)\mathit{dp}.$$E12

A sufficient condition for Eq. (12) to hold is obviously the “Orthogonality” of $\left(1/p\right)p_n\left(1/p\right)$ with respect to any “arbitrary” $\rho \left(1/p\right)$ (see Eq. (11)). The points $1/p_i$ are denoted by $1/{\left(p_i\right)}^n$ and they are the “zeros” of a certain set of “orthogonal polynomials” in the variable 1/p. By using these quantities in the “quadrature formula” we can obtain theoretically “exact accuracy” for “any polynomial” in 1/p up to the 16(th) degree.

A Fortran code is developed using NAG subroutines (The Numerical Algorithms Group (NAG)) [19].

The proposed technique will introduce details for all the interesting cases of a discontinuous cross section, as shown in Figure 1(a) and (b).

3. Calculation of the different inhomogeneous profiles

This section explains the proposed technique for calculating the dielectric profile for the two different inhomogeneous and complicated shapes of the cross section, as shown in Figure 1(a) and (b).

3.1 Calculation for circular dielectric material in the center of the cross section

The technique is based on Fourier transform and uses the image method and periodic replication for fulfilling the boundary conditions of the metallic waveguide. Periodicity and symmetry properties are chosen to force the boundary conditions at the location of the walls in a real problem, by extending the waveguide region ($0\le x\le a$, and $0\le y\le b$) to regions that are four-fold larger ($-a\le x\le a$, and $-b\le y\le b$). The elements of the matrix g(n,m) are calculated for an arbitrary profile in the cross section of the straight waveguide according to Figure 2(a) and (b).

Figure 2.

The image method for (a) an arbitrary profile in the cross section, and (b) the specific case of circular dielectric material in the center of the cross section.

The dielectric profile $g\left(x,y\right)$ is calculated according to $\epsilon \left(x,y\right)={\epsilon }_0\left(1+g\left(x,y\right)\right)$ and according to Figure 2(a) and (b) where $g\left(x,y\right)=g_0$. The specific case of circular dielectric material in the center of the cross section is shown in Figure 2(b) by using the image method. We obtain

$$\begin{array}{c}g\left(n,m\right)=\frac{g_0}{4\mathit{ab}}{\int }_{-a}^a\mathit{dx}{\int }_{-b}^{b}exp\left[-j\left(k_{x}x+k_{y}y\right)\right]\mathit{dy}\\ =\frac{g_0}{4\mathit{ab}}\left\{{\int }_{x_{11}}^{x_{12}}\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[-j\left(k_{x}x+k_{y}y\right)\right]\mathit{dy}+{\int }_{-x_{12}}^{-x_{11}}\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[-j\left(k_{x}x+k_{y}y\right)\right]\mathit{dy}\right.\\ +\left.{\int }_{-x_{12}}^{-x_{11}}\mathit{dx}{\int }_{-y_{12}}^{-y_{11}}exp\left[-j\left(k_{x}x+k_{y}y\right)\right]\mathit{dy}+{\int }_{x_{11}}^{x_{12}}\mathit{dx}{\int }_{-y_{12}}^{-y_{11}}exp\left[-j\left(k_{x}x+k_{y}y\right)\right]\mathit{dy}\right\}\\ =\frac{g_0}{4\mathit{ab}}\left\{{\int }_{x_{11}}^{x_{12}}\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[-j\left(k_{x}x+k_{y}y\right)\right]\mathit{dy}+{\int }_{x_{12}}^{x_{11}}-\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[-j\left(-k_{x}x+k_{y}y\right)\right]\mathit{dy}\right.\\ +\left.{\int }_{x_{12}}^{x_{11}}-\mathit{dx}{\int }_{y_{12}}^{y_{11}}-exp\left[-j\left(-k_{x}x-k_{y}y\right)\right]\mathit{dy}+{\int }_{x_{11}}^{x_{12}}\mathit{dx}{\int }_{y_{12}}^{y_{11}}-exp\left[-j\left(k_{x}x-k_{y}y\right)\right]\mathit{dy}\right\}\\ =\frac{g_0}{4\mathit{ab}}\left\{{\int }_{x_{11}}^{x_{12}}exp\left[j\left(k_{x}x\right)\right]\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[j\left(k_{y}y\right)\right]\mathit{dy}+{\int }_{x_{11}}^{x_{12}}exp\left[j\left(k_{x}x\right)\right]\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[-j\left(k_{y}y\right)\right]\mathit{dy}\right.\\ +\left.{\int }_{x_{11}}^{x_{12}}exp\left[-j\left(k_{x}x\right)\right]\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[j\left(k_{y}y\right)\right]\mathit{dy}+{\int }_{x_{11}}^{x_{12}}exp\left[-j\left(k_{x}x\right)\right]\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[-j\left(k_{y}y\right)\right]\mathit{dy}\right\}\\ =\frac{g_0}{4\mathit{ab}}\left\{{\int }_{x_{11}}^{x_{12}}\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[-j\left(k_{x}x+k_{y}y\right)\right]\mathit{dy}+{\int }_{x_{11}}^{x_{12}}\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[j\left(k_{x}x-k_{y}y\right)\right]\mathit{dy}\right.\\ +\left.{\int }_{x_{11}}^{x_{12}}\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[j\left(k_{x}x+k_{y}y\right)\right]\mathit{dy}+{\int }_{x_{11}}^{x_{12}}\mathit{dx}{\int }_{y_{11}}^{y_{12}}exp\left[-j\left(k_{x}x-k_{y}y\right)\right]\mathit{dy}\right\}\\ =\frac{g_0}{2\mathit{ab}}\left\{{\int }_{x_{11}}^{x_{12}}\left(exp\left({\mathit{jk}}_{x}x\right)+exp\left(-{\mathit{jk}}_{x}x\right)\right)\mathit{dx}{\int }_{y_{11}}^{y_{12}}cos\left(k_{y}y\right)\mathit{dy}\right\}.\end{array}$$E13

If $y_{11}$ and $y_{12}$ are functions of x, then we obtain

$$\begin{array}{c}g\left(n,m\right)=\frac{g_0}{{\mathit{abk}}_y}{\int }_{x_{11}}^{x_{12}}\left[sin\left(k_y{y}_{12}\left(x\right)\right)-sin\left(k_y{y}_{11}\left(x\right)\right)\right]cos\left(k_{x}x\right)\mathit{dx}\\ =\frac{2g_0}{\mathit{am\pi }}{\int }_{x_{11}}^{x_{12}}sin\left[\frac{\mathit{m\pi }}{2b}\left(y_{12}\left(x\right)-y_{11}\left(x\right)\right)\right]cos\left[\frac{\mathit{m\pi }}{2b}\left(y_{12}\left(x\right)+y_{11}\left(x\right)\right)\right]cos\left(\frac{\mathit{n\pi }}{a}x\right)\mathit{dx},\end{array}$$E14

where $k_x=\left(\mathit{n\pi }\right)/a$, and $k_y=\left(\mathit{m\pi }\right)/b$.

The radius of the circle is given by $r=\sqrt{{\left(x-a/2\right)}^2+{\left(y-b/2\right)}^2}$, thus for the specific case of the cross section (Figure 1(a)) and according to the image method, we obtain.

$$y_{11}\left(x\right)=b/2-\sqrt{r^2-{\left(x-a/2\right)}^2},$$E15

$$y_{12}\left(x\right)=b/2+\sqrt{r^2-{\left(x-a/2\right)}^2}$$E16

The dielectric profile for the cross section (Figure 1(a)) is given by.

$$g\left(n,m\ne 0\right)=\frac{2g_0}{\mathit{am\pi }}{\int }_{x_{11}}^{x_{12}}sin\left[\frac{\mathit{m\pi }}{2b}\left(y_{12}\left(x\right)-y_{11}\left(x\right)\right)\right]cos\left[\frac{\mathit{m\pi }}{2b}\left(y_{12}\left(x\right)+y_{11}\left(x\right)\right)\right]cos\left(\frac{\mathit{n\pi }}{a}x\right)\mathit{dx},$$E17

$$g\left(n,m=0\right)=\frac{g_0}{\mathit{ab}}{\int }_{x_{11}}^{x_{12}}\left(y_{12}\left(x\right)-y_{11}\left(x\right)\right)cos\left(\frac{\mathit{n\pi }}{a}x\right)\mathit{dx},$$E18

where $y_{12}\left(x\right)-y_{11}\left(x\right)=2\sqrt{r^2-{\left(x-a/2\right)}^2}$ and $y_{12}\left(x\right)+y_{11}\left(x\right)=b$.

The cyclic matrix $\mathbf{G}$ is given as follows. The Fourier transform is applied to the transverse dimension

$$\overline{g}\left(k_x,k_y\right)=F\left\{g\left(x,y\right)\right\}={\int }_x{\int }_{y}g\left(x,y\right)e^{-{\mathit{jk}}_{x}x-{\mathit{jk}}_{y}y}\mathit{\text{dxdy}}.$$E19

The components are organized in a vectorial notation as follows

$$\mathbf{E}=\left[\begin{array}{c}{\overline{E}}_{-N,-M}\\ ⋮\\ {\overline{E}}_{-N,+M}\\ ⋮\\ {\overline{E}}_{+n,+m}\\ ⋮\\ {\overline{E}}_{+N,+M}\end{array}\right].$$E20

The Fourier components of the dielectric profile are calculated in the Fourier space. The convolution operation

$${\overline{g}}^{\ast }\overline{E}=\left\{\sum _{n^{\prime }=-N}^N\sum _{m^{\prime }=-M}^M{g}_{n-n^{\prime },m-m^{\prime }}{E}_{n^{\prime },m^{\prime }}\right\}$$E21

is written in a matrix form as $\mathbf{G}E$ where

$$\overline{g}\left(n,m\right)\left(n^{\prime },m^{\prime }\right)=g_{n-n^{\prime },m-m^{\prime }}$$E22

and the matrix order is (2 N + 1)(2 M + 1), where $E$ is the electric field.

The convolution operation is expressed by the cyclic matrix $\mathbf{G}$ which consists of Fourier components of the dielectric profile ${\overline{g}}_{\mathit{nm}}$. Thus, the cyclic matrix $\mathbf{G}$ is given by the form

$$\mathbf{G}=\left[\begin{array}{ccccccc}{g}_{00}& g_{-10}& g_{-20}& \dots & g_{-\mathit{nm}}& \dots & g_{-\mathit{NM}}\\ g_{10}& g_{00}& g_{-10}& \dots & g_{-\left(n-1\right)m}& \dots & g_{-\left(N-1\right)M}\\ g_{20}& g_{10}& \ddots & \ddots & \ddots & & \\ ⋮& g_{20}& \ddots & \ddots & \ddots & & \\ g_{\mathit{nm}}& \ddots & \ddots & \ddots & g_{00}& ⋮& \\ ⋮& & & & & & \\ g_{\mathit{NM}}& \dots & \dots & \dots & \dots & \dots & g_{00}\end{array}\right].$$E23

The derivative of the dielectric profile is given by

$$g_x\left(n,m\right)=\frac{2g_0}{\mathit{am\pi }}{\int }_{x_{11}}^{x_{12}}sin\left[\frac{\mathit{m\pi }}{2b}\left(y_{12}\left(x\right)-y_{11}\left(x\right)\right)\right]cos\left[\frac{\mathit{m\pi }}{2b}\left(y_{12}\left(x\right)+y_{11}\left(x\right)\right)\right]cos\left(\frac{\mathit{n\pi }}{a}x\right)\mathit{dx},$$E24

where $y_{11}$ and $y_{12}$ are given according to Eqs (15) and (16). Similarly, we can calculate the value of $g_y\left(n,m\right)$, where $g_y\left(x,y\right)=\left(1/\epsilon \left(x,y\right)\right)\left(\mathit{d\epsilon }\left(x,y\right)/\mathit{dy}\right)$.

3.2 Calculation for the circular hollow core in the center of the cross section

Figure 3(a)–(c) shows the extending of the waveguide region in all cases to a four-fold larger region, according to the image method. The image method and periodic replication are needed for fulfilling the boundary condition of the metallic waveguide. Figure 3(a) shows the hollow waveguide where the circular hollow core is located in the center of the cross section. This figure represents an example of the complementary shape of Figure 3(c). Figure 3(b) shows the cross section entirely filled with the dielectric material. Figure 3(c) shows the cross section where the circular dielectric material is located in the center.

Figure 3.

Extending the waveguide region in all cases to a four-fold larger region, according to the image method. (a) The hollow waveguide where the circular hollow core is located in the center of the cross section. (b) The cross section entirely filled with dielectric material. (c) Circular dielectric material is located in the center of the cross section.

Note that the problem shown in Figure 3(a) is more complicated than the problem shown in Figure 3(c), and the technique for solving this inhomogeneous problem in the cross section based on the image method is not effective for the specific case shown in Figure 3(a). Thus the proposed technique for calculating the dielectric profile of this problem is based on the fact that this figure represents an example of the complementary shape of Figure 3(c).

In order to solve any inhomogeneous problem in the cross section (e.g., Figure 3(a) and (c)), more than one technique can be proposed for the same mode-model method.

The proposed technique to calculate the dielectric profile for the cross section as shown in Figure 3(a) for hollow waveguide is based on subtracting the dielectric profile of the waveguide with the dielectric material in the core (Figure 3(c)) from the dielectric profile of the waveguide filled entirely with the dielectric material (Figure 3(b)).

4. Numerical results

This section presents several examples for the different geometries of two specific examples of the complementary shapes of dielectric profile in the cross section, as shown in Figure 1(a)and(b). The solutions are demonstrated as a response to a half-sine ($\mathrm{T}{\mathrm{E}}_{10}$) input-wave profile.

A comparison with the known transcendental Equation [20] according to Figure 4(a) is given in order to examine the validity of the theoretical model. The known solution for the dielectric slab modes based on the transcendental Equation [20] is given as follows:

Figure 4.

(a) A dielectric slab in a rectangular metallic waveguide. (b) A comparison between the theoretical model (Eq. (2)) and the transcendental equation (Eqs (25)–(27)) according to Ref. [20], where a = 2b = 2 cm, d = 3.3 mm, ${\epsilon }_r$= 9, and $\lambda$= 6.9 cm. (c) The convergence of our theoretical results.

$$E_{y1}=j\frac{k_z}{{\epsilon }_0}sin\left(\mathit{\nu x}\right)0<x<t$$E25

$$E_{y2}=j\frac{k_z}{{\epsilon }_0}\frac{sin\left(\mathit{\nu t}\right)}{cos\left(\mu \left(t-a/2\right)\right)}cos\left[\mu \left(x-a/2\right)\right]t<x<t+d$$E26

$$E_{y3}=j\frac{k_z}{{\epsilon }_0}sin\left[\nu \left(a-x\right)\right]t+d<x<a,$$E27

where $\nu \equiv \sqrt{k_o^2-k_z^2}$ and $\mu \equiv \sqrt{{\epsilon }_r{k}_o^2-k_z^2}$ result from the transcendental equation

$$\left(\frac{a-d}{d}\right)\frac{\mathit{d\mu }}{2}tan\left(\frac{\mathit{d\mu }}{2}\right)-\left(\mathit{t\nu }\right)cot\left(\mathit{t\nu }\right)=0.$$E28

The solution obtained for the wave profile ((25)–(27)) describes a symmetrical mode of the dielectric slab. This mode is substituted as an input wave at z = 0 to the solution of the proposed theoretical model Eq. (2).

The comparison between the theoretical model (Eq. (2)) and the transcendental equation (Eqs (25)–(27)) is shown in Figure 4(b) for the dielectric slab in a rectangular metallic waveguide (Figure 4(a)) and the convergence of our theoretical results is shown in Figure 4(c).

The comparison is demonstrated for every order (N = 1, 3, 5, 7, and 9). The order $N$ determines the accuracy of the solution. The convergence of the solution is verified by the criterion for the $E_y$ component of the fields.

The convergence of the solution is verified by the criterion

$$C\left(N\right)\equiv \mathit{log}\left\{\frac{max\left(|E_y^{N+2}-E_y^N|\right)}{\mid max\left(E_y^{N+2}\right)-min\left(E_y^N\right)\mid }\right\},N\ge 1.$$E29

where the number of the modes is equal to ${\left(2N+1\right)}^2$. The order $N$ determines the accuracy of the solution.

If the value of the criterion (Eq. (29)) is less than −2, then the numerical solution is well converged. When N increases, then $E_y\left(N\right)$ approaches $E_y$. The value of the criterion between N = 7 and N = 9 is equal to −2.38 $\simeq$ −2, namely a hundredth part. Comparison between the theoretical mode-model (Eq. (2)) and the known model [20] shows good agreement.

Figure 5(a)–(e) shows the output field where the circular dielectric material is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, ${\epsilon }_r$ = 3, 5, 7, 9, for r = 2.5 mm, where r is the radius of the circular dielectric material. The output field in the same cross section of the results Figure 5(a)–(d) are shown in Figure 5(e) for the x-axis and where y = b/2 = 10 mm, for the values of ${\epsilon }_r$ = 3, 5, 7, and 9, respectively. Figure 6(a)–(e) demonstrates the output fields by changing only the parameter of the radius of the circular dielectric material from r = 2.5 to r = 2. The other parameters are a = b = 20 mm, $k_0$ = 167 $1/\mathrm{m}$, $\lambda$ = 3.75 cm, and $\beta$ = 58 $1/\mathrm{m}$.

Figure 5.

The output field where the circular dielectric material is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, and r = 2.5 mm, r is the radius of the circular dielectric material, and for (a) ${\epsilon }_r$ = 3, for (b) ${\epsilon }_r$ = 5, for (c) ${\epsilon }_r$ = 7, and for (d) ${\epsilon }_r$ = 9. (e) The output field in the same cross section of the results (a)–(d) for x-axis and where y = b/2 = 10 mm, for the values of ${\epsilon }_r$ = 3, 5, 7, and 9, respectively. The other parameters are a = b = 20 mm, $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, and β =58 1/m.

Figure 6.

The output field where the circular dielectric material is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, and r = 2 mm, where r is the radius of the circular dielectric material, and for (a) ${\epsilon }_r$ = 3, for (b) ${\epsilon }_r$ = 5, for (c) ${\epsilon }_r$ = 7, and for (d) ${\epsilon }_r$ = 9. (e) The output field in the same cross section of the results (a)–(d) for x-axis and where y = b/2 = 10 mm, for the values of ${\epsilon }_r$ = 3, 5, 7, and 9, respectively. The other parameters are a = b = 20 mm, $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, and β = 58 1/m.

Figure 7(a)–(e) shows the output field where the circular hollow core is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, ${\epsilon }_r$ = 1.5, 1.6, 1.7, 1.8, for r = 2.5 mm, where r is the radius of the circular hollow core. The output field in the same cross section of the results Figure 7(a)–(d) are shown in Figure 7(e) for x-axis and where y = b/2 = 10 mm, for the values of ${\epsilon }_r$ = 1.5, 1.6, 1.7, and 1.8, respectively. Figure 8(a)–(e) demonstrates the output fields by changing only the parameter of the radius of the circular hollow core from r = 2.5 to r = 2. The other parameters are a = b = 20 mm, $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, and $\beta$ = 58 1/m.

Figure 7.

The output field where the circular hollow core is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, and r = 2.5 mm, where r is the radius of the circular hollow core, and for (a) ${\epsilon }_r$ = 1.5, for (b) ${\epsilon }_r$ = 1.6, for (c) ${\epsilon }_r$ = 1.7, and for (d) ${\epsilon }_r$ = 1.8. (e) The output field in the same cross section of the results (a)–(d) for x-axis and where y = b/2 = 10 mm, for the values of ${\epsilon }_r$ = 1.5, 1.6, 1.7, and 1.8, respectively. The other parameters are a = b = 20 mm, $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, and β = 58 1/m.

Figure 8.

The output field where the circular hollow core is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, and r = 2 mm, where r is the radius of the circular hollow core, and for (a) ${\epsilon }_r$ = 1.5, for (b) ${\epsilon }_r$ = 1.6, for (c) ${\epsilon }_r$ = 1.7, and for (d) ${\epsilon }_r$ = 1.8. (e) The output field in the same cross section of the results (a)–(d) for x-axis and where y = b/2 = 10 mm, for the values of ${\epsilon }_r$ = 1.5, 1.6, 1.7, and 1.8, respectively. The other parameters are a = b = 20 mm, $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, and β =58 1/m.

By increasing only the dielectric constant from ${\epsilon }_r$ = 3 to ${\epsilon }_r$ = 9, according to Figures 5(a)–(e) and 6(a)–(e), and from ${\epsilon }_r$ = 1.5 to ${\epsilon }_r$ = 1.8, according to Figures 7(a)–(e) and 8(a)–(e), the Gaussian shape of the output transverse profile of the field increased, the $\mathrm{T}{\mathrm{E}}_{10}$ wave profile decreased, and the relative amplitude of the output field decreased.

We can predict the waveguide parameters (${\epsilon }_r$ and r) for obtaining the Gaussian behavior of the output field in all case. The cross section in the first interesting case consists of circular dielectric material in the center of the cross section (Figure 1(a)). The cross section in the second interesting case consists of a circular hollow core in the center of the cross section (Figure 1(b)). The output results refer to the same parameters a = b = 20 mm, $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, and $\beta$ = 58 1/m. According to the results of the first case, in order to obtain the Gaussian behavior, the values of ${\epsilon }_r$ = 3, 5, 7, 9 and r = 2 or r = 2.5 are needed. In the second case, in order to obtain the Gaussian behavior, the values of ${\epsilon }_r$ = 1.5, 1.6, 1.7, and 1.8 and r = 2 or r = 2.5 are needed.

These results are strongly affected by the different parameters ${\epsilon }_r$ and r, and for the same other parameters of $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, $\beta$ = 58 1/m, and the dimensions of the rectangular cross section.

5. Conclusions

The wavelet transform creates a representation of the signal in both the time and frequency domain in order to allow efficient access of localized information about the signal. A set of waveforms comprising a transform is called a basis function. Fourier transforms use only sine and cosine waves as their basic functions, namely a signal is decomposed into a series of sine and cosine functions or wavelets by the FFT. Examples for the applications of wavelet transform are demonstrated in [2, 3, 4, 5]. The proposed method in this chapter is based on the Fourier transform that creates a representation of the signal in the frequency domain.

Two specific examples of complementary shapes of dielectric profile in the cross section were introduced in this chapter. In the first case, the cross section consists of circular dielectric material in the center of the cross section. In the second case, the cross section shows the complementary shape of the cross section of the first case, as an example of a hollow waveguide in which the circular hollow core is located in the center of the cross section.

Note that the problem shown in Figure 3(a) is more complicated than the problem shown in Figure 3(c), and the technique for solving this inhomogeneous problem in the cross section based on the image method is not effective for the specific case shown in Figure 3(a). The proposed technique for calculating the dielectric profile of the problem shown in Figure 3(a) is based on the fact that this figure represents an example of the complementary shape of Figure 3(c).

In order to solve any inhomogeneous problem in the cross section (e.g., Figure 3(a) and (c)), more than one technique can be proposed for the same mode-model method.

The proposed technique to calculate the dielectric profile for the cross section as shown in Figure 3(a) for hollow waveguide is based on subtracting the dielectric profile of the waveguide from the dielectric material in the core (Figure 3(c)) from the dielectric profile of the waveguide filled entirely with the dielectric material (Figure 3(b)).

Figures 5(a)–(e) and 6(a)–(e) demonstrate the output fields, where the circular dielectric material is located in the center of the cross section of the straight rectangular waveguide, where the parameter r refers to the radius of the circular dielectric material. Figures 7(a)–(e) and 8(a)–(e) demonstrate the output fields, where the circular hollow core is located in the center of the cross section of the straight rectangular waveguide, where the parameter r refers to the radius of the circular hollow core. The other parameters are a = b = 20 mm, $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, and $\beta$ = 58 1/m.

By increasing only the dielectric constant from ${\epsilon }_r$ = 3 to ${\epsilon }_r$ = 9, according to Figures 5(a)–(e) and 6(a)–(e), and from ${\epsilon }_r$ = 1.5 to ${\epsilon }_r$ = 1.8, according to Figures 7(a)–(e) and 8(a)–(e), the Gaussian shape of the output transverse profile of the field increased, the $\mathrm{T}{\mathrm{E}}_{10}$ wave profile decreased, and the relative amplitude of the output field decreased.

We can predict the waveguide parameters (${\epsilon }_r$ and r) for obtaining the Gaussian behavior of the output field in all cases. The output results refer to the same parameters a = b = 20 mm, $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, and $\beta$ = 58 1/m. According to the results of the first case, in order to obtain the Gaussian behavior, the values of ${\epsilon }_r$ = 3, 5, 7, 9 and r = 2 or r = 2.5 are needed. In the second case, in order to obtain the Gaussian behavior, the values of ${\epsilon }_r$ = 1.5, 1.6, 1.7, and 1.8 and r = 2 or r = 2.5 are needed.

The results are strongly affected by the different parameters ${\epsilon }_r$ and r, and for the same other parameters of $k_0$ = 167 1/m, $\lambda$ = 3.75 cm, $\beta$ = 58 1/m, and the dimensions of the rectangular cross section.

The applications are useful for straight rectangular waveguides in millimeter regimes, where the circular dielectric material is located in the center of the cross section, and also for hollow waveguides, where the circular hollow core is located in the center of the cross section.

The wavelet transform creates a representation of the signal in both the time and frequency domain in order to allow efficient access of localized information about the signal. A set of waveforms comprising a transform is called a basis function. Fourier transforms use only sine and cosine waves as its basic functions, namely a signal is decomposed into a series sine and cosine functions or wavelets by the FFT. Examples for the applications of wavelet transform are demonstrated in [2, 3, 4, 5]. The proposed method in this chapter is based on the Fourier transform that creates a representation of the signal in the frequency domain. The main points of the proposed method and the proposed technique are:

1. A Laplace transform is necessary to obtain convenient and simple input–output connections of the fields. The method consists of Fourier coefficients of the transverse dielectric profile and of the input–output profile. Thus, the accuracy of the method depends on the number of the modes in the system.

2. The Laplace transform

   $$\stackrel{\sim }{a}\left(s\right)=L\left\{a\left(\zeta \right)\right\}={\int }_{\zeta =0}^{\infty }a\left(\zeta \right)e^{-\mathit{s\zeta }}\mathit{d\zeta },$$E30

   is applied on the z-dimension, where $a\left(z\right)$ represents any z-dependent variables of the wave equations.

3. A Fourier transform is applied on the transverse dimension

   $$\overline{g}\left(k_x,k_y\right)=F\left\{g\left(x,y\right)\right\}={\int }_x{\int }_{y}g\left(x,y\right)e^{-{\mathit{jk}}_{x}x-{\mathit{jk}}_{y}y}\mathit{\text{dxdy}},$$E31

   and the differential equations are transformed to an algebraic form in the ($\omega ,s,k_x,k_y$) space.

4. The method of images is applied to satisfy the conditions $\widehat{n}\times E=0$ and $\widehat{n}\cdot \left(\nabla \times E\right)=0$ on the surface of the ideal metallic waveguide walls, where $\widehat{n}$ is a unit vector perpendicular to the surface. The dielectric profile, $g\left(x,y\right)$, is defined inside the waveguide boundaries, $0\le x\le a$ and $0\le y\le b$. In order to maintain the boundary conditions without physical metallic walls, a substitute physical problem is constructed with infinite transverse extent. The periodicity and the symmetry properties are chosen to force the boundary conditions at the location of the walls in the real problem. This is done by extending the waveguide region $0\le x\le a,0\le y\le b$ to a four-fold larger region. Hence, the following relations are yielded

   $$g\left(-x,y\right)=g\left(x,-y\right)=g\left(x,y\right)=g\left(-x,-y\right),$$E32
   $$E_x\left(x,-y\right)=-E_x\left(x,y\right),E_x\left(-x,y\right)=E_x\left(x,y\right).$$E33

   The region $-a\le x\le a,-b\le y\le b$ is then further extended to infinity by periodic replication, $g\left(x+2\mathrm{\ell }a,y+2\mathit{kb}\right)=g\left(x,y\right)$, where $-\infty <\mathrm{\ell },k<\infty$. The field components are periodically, namely, $E_x\left(x+2\mathrm{\ell }a,y+2\mathit{kb}\right)=E_x\left(x,y\right)$ for $-\infty <\mathrm{\ell },k<\infty$. The substitution of the physical problem is equivalent to the original problem in the region $0\le x\le a$, $0\le y\le b$, and satisfies the same boundary conditions on the boundary of this region. The discrete Fourier transform series is given with $k_x=\mathit{n\pi }/a$ and $k_y=\mathit{m\pi }/b$, and the transverse wavenumbers are given by $k_{\mathit{ox}}=\pi /a$, and $k_{\mathit{oy}}=\pi /b$, where $a$ and $b$ are the transverse dimensions of the rectangular boundaries. We substitute $k_x={\mathit{nk}}_{\mathit{ox}}$ and $k_y={\mathit{mk}}_{\mathit{oy}},$ where the integers n and m are truncated by $-N\le n\le N$ and $-M\le m\le M$, respectively. The orders $N$ and $M$ determine the accuracy of the solution.

5. The output transverse field profiles are given by the inverse Laplace and Fourier transforms, as follows

   $${\mathbf{E}}_{\mathbf{y}}\left(\mathbf{x},\mathbf{y},\mathbf{z}\right)=\sum _{\mathbf{n}}\sum _{\mathbf{m}}{\int }_{\sigma -{\mathbf{j}}_{\infty }}^{\sigma +{\mathbf{j}}_{\infty }}{\mathbf{E}}_{\mathbf{y}}\left(\mathbf{n},\mathbf{m},\mathbf{s}\right)exp\left[j{\mathbf{nk}}_{\mathbf{ox}}\mathbf{x}+{\mathbf{jmk}}_{\mathbf{oy}}\mathbf{y}+\mathbf{sz}\right]\mathbf{ds},$$E34

   where the inverse Laplace transform is calculated according to the Salzer method [17, 18].

6. The main objective is to generalize the mode model method [16] in order to solve also complicated and practical problems of circular dielectic material and a circular hollow core (Figure 1(a) and (b)) in the center of the cross section of the rectangular waveguide.

7. The second objective is to find the relevant parameters in order to obtain the Gaussian behavior of the output field in the interesting cases of circular dielectic material and a circular hollow core in the rectangular cross section.