## Abstract

This chapter presents two interesting types of dielectric materials in the straight rectangular waveguides. Five examples of the different discontinuous cross sections and complementary shapes will demonstrate. We will introduce in all case the effective technique to calculate the dielectric profile in the cross section. The first type will demonstrate where the dielectric material is located in the center of the cross section. The second type will demonstrate where the hollow core is located in the center of the cross section in the case of the hollow waveguide. The two different types are complementary shapes for two different applications. The proposed techniques relate to the method based on Laplace and Fourier transforms and the inverse Laplace and Fourier transforms. The method is based also on Fourier transform, thus we need use with the image method to calculate the dielectric profile in the cross section. The image method and periodic replication are needed for fulfilling the boundary condition of the metallic waveguide. The applications are useful for straight waveguides in millimeter regimes, in the cases where the dielectric profile is located in the center of the cross section, for cases where the hollow rectangle is located in the center of the cross section, and also for complicated and discontinuous profiles in the cross section.

### Keywords

- wave propagation
- dielectric profiles
- rectangular waveguides
- dielectric materials

## 1. Introduction

Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides with emphasis on recent developments has been published [1]. In this review, interesting methods are given such as, the finite-difference and the finite-element methods. Review of numerical methods for the analysis of the homogeneous and inhomogeneous, isotropic and anisotropic, microwave and optical dielectric waveguides with arbitrarily-shaped cross sections has been published [2]. The main approaches as the integral equations, finite difference, and finite element have been discussed.

A fundamental and accurate approach to compute the attenuation of electromagnetic waves propagating has been proposed [3]. The propagation constant was found by substituting the values of transverse wave numbers into the dispersion relation. A Green’s dyadic for describing the propagating electromagnetic waves in a rectangular dielectric waveguide has been developed [4]. The use of Green’s dyadic developed in order to calculate the effect of small perturbations upon the system.

An analytical method for solution of one-dimensional optical systems, based on the differential transfer matrices has been presented [5]. An approach to solve the problem of the propagation of electromagnetic waves in unidimensional media with an arbitrary variation of their dielectric permittivity has been proposed [6]. This method was deduced from the Maxwell equations with a minimum of approximations and allows a full vectorial description of both the electric and magnetic fields through the direct calculation of their cartesian coordinates. The equations permit the simulation of materials with a continuous variation of their dielectric permittivity without approximating them by discontinuous layered media, reducing so the computational effort of the models.

An analytical method for solution of non-homogeneous anisotropic optical systems, based on the extension of transfer matrices into differential form has been presented [7]. This approach can be used for exact calculation of various functions including reflection and transmission coefficients, band structures and bound states. A full-wave analysis of lossy dielectric waveguides using a hybrid vector finite element method has been presented [8]. The direct matrix solution technique with minimum degree of reordering has been combined with the modified Lanczos algorithm to solve for the resultant sparse generalized eigenmatrix equation efficiently.

Three-dimensional finite-element method with edge elements for electromagnetic waveguide discontinuities has been proposed [9]. This paper shows that the finite-element method using edge elements succeeds in suppressing spurious solutions and moreover that this method succeeds in the analysis of three-dimensional electromagnetic waveguide problems with metal wedges. An analytical approach based on scalar wave approximation to estimate the modal dispersion characteristics and cutoff condition of an optical waveguide has been presented [10]. This approach has an arbitrary and uniform core cross-section. The structure represents the core of a circular waveguide which is compressed at both the ends of a diameter.

Propagation characteristics of modes in some rectangular waveguides using the finite-difference time-domain method have been studied [11]. The method in this paper was used to determine the modal characterization of rectangular waveguide structures by means of a least-square non-linear fitting to a theoretical modal expansion. Wave propagation along a rectangular waveguide with slowly varying width has been investigated [12] with the help of field theory and approximate circuit theory. Many properties of the modulated periodic structure, e.g., the frequency dependence of the propagation constant, group and phase velocities, and the electric field axial variation for the fundamental space harmonic and its filter-like property have been investigated.

A method for measuring samples using a partially-filled waveguide has been presented [13]. A mode-matching technique was used to determine the fields in the three regions. An advantageous finite element method for the rectangular waveguide problem has been developed [14] by which complex propagation characteristics may be obtained for arbitrarily shaped waveguide.

A method has been introduced for the frequency domain analysis of arbitrary longitudinally inhomogeneous waveguides [15]. The integral equations of the longitudinally inhomogeneous waveguides, converted from their differential equations, were solved using the method of moments. A method has been introduced for the frequency domain analysis of arbitrarily loaded lossy and dispersive nonuniform transmission lines [16]. In this method, all distributed primary parameters of the line and also the voltage and current distribution along the line were considered as a Taylor’s series.

A technique based on the two-dimensional Fourier transform has been presented [17] and applied to the study of nonlinear wave propagation phenomena in one-dimensional, finite, nonlinear transmission lines.

A method for calculating the modes of arbitrarily shaped dielectric waveguides has been presented [18]. It consists of expanding the field in a two-dimensional Fourier series. The expansion has been used to convert the scalar wave equation into a matrix eigenvalue equation. A transfer matrix function for the analysis of electromagnetic wave propagation along the straight dielectric waveguide with arbitrary profiles has been proposed [19]. This method is based on the Laplace and Fourier transforms and the inverse Laplace and Fourier transforms.

## 2. Complicated and discontinuous profiles in the cross section

In this chapter, we present some examples of dielectric structures as shown in Figure 1a–e. The method is based on Fourier transform, thus we need use with the image method and periodic replication for fulfilling the boundary conditions of the metallic waveguide. We relate also to the complementary shapes for different applications. The periodicity and the symmetry properties are chosen to force the boundary conditions at the location of the walls in real problem, by extending the waveguide region (

Note that the geometries from Figure 1a–e become more complex. Calculating of the dielectric profile for Figure 1a is the simplest and for Figure 1e is the most complicated in relation to the examples described in these examples. The integrals of the dielectric profiles of the geometries described in Figure 1a and b can be calculated by analytical form and without solving numerical form. The calculation of the dielectric profile for the geometry of Figure 1c for a hollow rectangle in the center is interesting in the case of hollow waveguide, but it is more complicated than the case of Figure 1b. Therefore, in this case we should calculate the dielectric profile by substracting the dielectric profile of Figure 1b from the dielectric profile of Figure 1a. The calculation of the integrals of the dielectric profiles for the geometries depicted in Figure 1d and e are already more complicated, and in such cases the proposed techniques require that the integrals be solved numerically. We will explain the proposed technique for calculating the dielectric profile for each case.

Figure 2c represents an example of receiving a hollow waveguide where the hollow rectangle is located in the center by substracting the waveguide with dielectric material located at the center of the cross section (Figure 2b) from the waveguide entirely filled with the dielectric material (Figure 2a), and by using the image method (Figure 2d). Figure 3c represents an example of receiving a hollow waveguide with four hollow rectangles by substracting the four dielectric rectangles (Figure 3b) from the waveguide entirely filled with the dielectric material (Figure 3a), and by using with the image method (Figure 3d).

## 3. The analytical technique to calculate the dielectric profiles

This section presents a technique to calculate the dielectric profile for the two inhomogeneous geometries of the cross section, as shown in Figure 4a and b.

Figure 4a and b show the cross section of Figure 1b and c, respectively, with the relevant parameters. These figures represent two examples of the complementary shapes in the cross section. The periodic replication is shown in Figure 4c by using with the image method. The dielectric material in Figure 4a is located in the center of the cross section. The hollow rectangle in Figure 4b is located in the center of the cross section in the case of the hollow waveguide.

The dielectric profile

where

For the cross section as shown in Figure 4a and according to Figure 4c, the center of the dielectric rectangle is located at (a/2, b/2). According to Figure 4a and c, the Fourier components of the dielectric profile are given by

Similarly, for the cross section entirely filled with the dielectric material as shown in Figure 1a, the Fourier components of the dielectric profile are given by

The Fourier components of the dielectric profile (

## 4. The numerical technique to calculate the dielectric profiles

### 4.1 Calculation of the dielectric profile according to the ω ε function

Figure 5a and b show the cross section of Figure 1d and e, respectively, with the relevant parameters. These figures represent two examples of the complementary shapes in the cross section. The cross section consists with four dielectric profiles as shown in Figure 5a. The hollow waveguide with four hollow rectangles is shown in Figure 5b. The centers of the first, the second, the third and the firth dielectric rectangles in Figure 5a and hollow rectangles in Figure 5b are located at the points (a/4, b/4), (3a/4, b/4), (a/4, 3b/4), and (3a/4, 3b/4), respectively. The calculation of the elements of the matrix in Figure 5b is already more complicated. Thus, the Fourier components of the dielectric profile (

In order to calculate the elements of the dielectric profiles of the inhomogeneous geometry of the cross section in Figure 5a, we can use with the

### 4.2 The technique based on ω ε function for the inhomogeneous cross section

It is good idea to calculate the dielectric profile of Figure 4a again, by using the

and

where

Similarly, we can calculate the rectangular dielectric profile according to the location of the profile in the cross section of the waveguide.

The elements of the matrix in Fourier space of the inhomogeneous geometry described in Figure 4a are given in the case of b

### 4.3 The technique based on ω ε function for the inhomogeneous cross section

Figure 5a shows the cross section where the centers of the first, the second, the third and the firth rectangles are located at the points (a/4, b/4), (3a/4, b/4), (a/4, 3b/4), and (3a/4, 3b/4), respectively, and for

The dielectric profile of Figure 5a is given according to the _{ε} function by

and

where

The elements of the matrix for the inhomogeneous geometry of the cross section of Figure 5a are calculated in Fourier space by

According to Ref. [19]. The cyclic matrix * G* is given by the form

Similarly, the

The proposed technique to calculate the elements of the matrix

## 5. Numerical results

This section presents several examples for the different geometries in the cross section. Five examples of the different discontinuous cross sections and complementary shapes are demonstrated in Figure 1a–e. All the next graphical results are demonstrated as a response to a half-sine (

Figure 7a–e shows the results of the output field as a response to a half-sine (

The results of Figure 7a–e are demonstrated for

The output field for the hollow rectangular waveguide where the dielectric material is located between the hollow rectangle and the metal is shown in Figure 8a–d, where a = b = 20 mm, and c = d = 14 mm. In this case the thickness of the dielectric layer is equal to 3 mm (e = f = 3 mm). The results are demonstrated for

We can find the relevant parameters to obtain the Gaussian behavior of the output field. We can show that the Gaussian behavior is obtained when the thickness of the dielectric layer is equal to 3 mm, as shown in the results of Figure 8a–d. The output results are strongly affected by the thickness of the dielectric layer.

Figure 9a–d shows the results of the output field as a response to a half-sine (

Figure 10a–d shows the results of the output field as a response to a half-sine (

By increasing only the parameter

The applications are useful for straight waveguides in millimeter regimes, in the cases where the dielectric profile is located in the center of the cross section, for the cases where the hollow rectangle is located in the center of the cross section, and also for complicated and discontinuous profiles in the cross section.

## 6. Conclusions

This chapter presents two interesting types of dielectric materials in the straight rectangular waveguides and their applications. Five examples of the different discontinuous cross section were demonstrated. The effective technique was proposed for all case of Figure 1a–e, in order to calculate the dielectric material in the specific cross section.

The proposed techniques are very important to understand the influence of the dielectric materials and the hollow rectangles in all case of discontinuous geometry in the cross section. All the graphical results are demonstrated as a response to a half-sine (

The method is based on Fourier transform, thus we need use with the image method to calculate the dielectric profile in the cross section. The image method and periodic replication are needed for fulfilling the boundary condition of the metallic waveguide.

Figure 8a–e relates to the hollow rectangular waveguide where the hollow rectangle is located in the center of the cross section as shown in Figure 4b. We can find the relevant parameters to obtain the Gaussian behavior of the output field. From the results of Figure 8a–e, the Gaussian behavior is obtained when the thickness of the dielectric layer is equal to 3 mm. The output results are strongly affected by the thickness of the dielectric layer.

By increasing only the parameter

The applications are useful for straight waveguides in millimeter regimes, in the cases where the dielectric profile is located in the center of the cross section, for cases where the hollow rectangle is located in the center of the cross section, and also for complicated and discontinuous profiles in the cross section.

## References

- 1.
Chiang KS. Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides. Optical and Quantum Electronics. 1994; 26 :S113-S134 - 2.
Saad M. Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides. IEEE Transactions on Microwave Theory and Techniques. 1985; MTT-33 :894-899 - 3.
Yeap KH, Teh KH, Yeong KC, Lai KC, Loh MC. Propagation in dielectric rectangular waveguides. Optica Applicata. 2016; XLVI :317-330 - 4.
Bartling JQ. Propagation of an electromagnetic wave in an infinite rectangular dielectric wave guide. Journal of the Franklin Institute. 1969; 287 :389-407 - 5.
Khorasani S. Differential transfer-matrix method for solution of one-dimensional linear nonhomogeneous optical structures. Optical Society of America. 2003; 20 :91-96 - 6.
Rojas JAM, Alpuente J, Pineiro J, Sanchez R. Rigorous full vectorial analysis of electromagnetic wave propagation in 1D inhomogeneous media. Progress in Electromagnetics Research. 2006; 63 :89-105 - 7.
Mehrany K, Khorasani S. Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices. Journal of Optics A. 2002; 4 :624-635 - 8.
Lee JF. Finite element analysis of lossy dielectric waveguides. IEEE Transactions on Microwave Theory and Techniques. 1994; 42 :1025-1031 - 9.
Ise K, Inoue K, Koshiba M. Three-dimensional finite-element method with edge elements for electromagnetic waveguide discontinuities. IEEE Transactions on Microwave Theory and Techniques. 1991; 39 :1289-1295 - 10.
Singh V, Joshi M, Prasad B, Ojha P. Modal dispersion characteristics and waveguide dispersion of an optical waveguide having a new unconventional core cross-section. Journal of Electromagnetic Waves and Applications. 2004; 18 :455-468 - 11.
Hernandez-Lopez MA, Quintillan M. Propagation characteristics of modes in some rectangular waveguides using the finite-difference time-domain method. Journal of Electromagnetic Waves and Applications. 2000; 14 :1707-1722 - 12.
Mallick AK, Sanyal GS. Electromagnetic wave propagation in a rectangular waveguide with sinusoidally varying width. IEEE Transactions on Microwave Theory and Techniques. 1978; MTT-26 :243-249 - 13.
Bogle A, Havrilla M, Nyquis D, Kempel L, Rothwell E. Electromagnetic material characterization using a partially-filled rectangular waveguide. Journal of Electromagnetic Waves and Applications. 2005; 19 :1291-1306 - 14.
Vaish A, Parthasarathy H. Analysis of a rectangular waveguide using finite element method. Progress in Electromagnetics Research C. 2008; 2 :117-125 - 15.
Khalaj-Amirhosseini M. Analysis of longitudinally inhomogeneous waveguides using the method of moments. Progress In Electromagnetics Research. 2007; 74 :57-67 - 16.
Khalaj-Amirhosseini M. Analysis of nonuniform transmission lines using Taylor’s series expansion. International Journal of RF and Microwave Computer-Aided Engineering. 2006; 10 :536-544 - 17.
Molina-Fernandez I, Camacho-Penalosa C, Ramos JI. Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena. IEEE Transactions on Microwave Theory and Techniques. 1994; 42 :1079-1085 - 18.
Henry CH, Verbeek BH. Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis. Journal of Lightwave Technology. 1989; 7 :308-313 - 19.
Menachem Z, Jerby E. Transfer matrix function (TMF) for propagation in dielectric waveguides with arbitrary transverse profiles. IEEE Transactions on Microwave Theory and Techniques. 1998; 46 :975-982 - 20.
Vladimirov V. Equations of Mathematical Physics. New York, NY: Marcel Dekker; 1971