**2.1. Analytical models: circular and elliptical dielectric tunnels**

for communication channel characterization. In this paper, we shall focus on tunnel environments. The prediction and characterization of the radio-wave propagation is needed

Confined environments are complex real-life electromagnetic (EM) problems. Several techniques and methods are used to study the radio-wave propagation or to design antennas to achieve some good performance. Experimental techniques, analytical methods (exact solutions), numerical methods (approximate solutions), asymptotic methods (approximate high-frequency expansions of Maxwell's equations), and approximate techniques (approximate solutions applicable for certain types of electromagnetic problems) are among the most commonly used techniques in confined structures. Experiments might be expensive and time consuming. Analytical and approximate techniques are limited to some structures. In turn, asymptotic methods are mostly used to study the wave propagation in tunnels. However, in many cases, the antennas employed to provide the communication in these systems are strongly affected by its surrounding environment, affecting the performance of the system. Thus, near-field considerations have to be accounted for, which cannot be considered by these techniques. Finally, the practical utilization of numerical full-wave methods has been hampered by their large computational time

With the increasing development of computers, appropriate new models and simplifications are being developed. The formulation of an efficient modal approach stemming from the fact that tunnels can be modeled by an over-sized waveguide and the *a priori* knowledge of the fields in the axial direction is presented. It allows one to have a better physical insight into wave propagation in confined environments, as well as dealing with electrically large structures like tunnels. The mode parameter determination is carried out by a full-wave

The calculation volume has to be limited in full-wave volumic methods, and we are interested only in the fields inside the tunnel. Thus, the electromagnetic modeling of the tunnel walls, which can be lossy dielectric, is addressed. Lastly, by assimilating a confined environment to a lossy dielectric waveguides of arbitrary cross section, the mode extraction of these structures is presented. To the best of the authors' knowledge, no such model has been

The chapter is structured as follows: In the first section, an overview of the principal techniques for the description of the EM wave propagation in above structures is briefly presented. In the second section, the formulation of the modal approach is described in detail. In the following section, the implementation for a simple canonical case is shown. Lastly, the numerical analysis of multimodal waveguides representative of confined environments is illustrated in a realistic rectangular tunnel. Finally, discussions and

**2. Modeling approaches for wave propagation in confined environments** The correct understanding of wave propagation in confined structures like tunnels has been an important area of research and development. They have been studied in the last 40 years for radio communication system deployment. Unfortunately, as we shall see, current models cannot be generally applied or they do not describe adequately the wave propagation. The

time-domain method, namely, the transmission-line matrix (TLM) method.

conclusions are developed at the end of the chapter.

to optimize the system performances.

240 Advanced Electromagnetic Waves

compared to asymptotic methods.

reported.

No analytic solution exists for the problem of wave propagation along a dielectric waveguide of arbitrary cross-sectional shape. The circular and elliptical waveguides are the only cases for which analytic solutions for guided waves exist. Field solution in waveguides for analytical and approximate methods is usually expressed in terms of modes. They are classified into different types according to their field configuration: transverse electric modes (TE), with no electric field in the direction of propagation; transverse magnetic modes (TM), with no magnetic field in this direction; transverse electromagnetic modes (TEM), with neither electric nor magnetic field in this direction; and hybrid modes (HE) or (EH), with nonzero electric and magnetic fields in the direction of propagation.

Dielectric circular waveguides can support a family of circularly symmetric *TEnm* or *TMnm* modes, with *n* = *m* = 0, and hybrid modes *HEnm* and *EHnm*. The subscripts *n* and *m* denote the number of oscillations with the cylindrical coordinates *ρ* and *φ*, respectively. Some discussions considering different excitations, dependence on the constitutive parameters of the tunnel walls, and other interesting topics are treated in [9]. The details of the determination of the fields will not be repeated here. Finally, the wave propagation in elliptical tunnels has not been treated. However, a deformed circular waveguide can be approximated by an elliptical cylinder. Detailed theoretical as well as experimental results can be found in [34].

### **2.2. Approximate models: rectangular dielectric tunnels**

An exact analytic solution does not exist for the case of wave propagation in a rectangular tunnel with lossy dielectric walls. An approximate approach, namely, Marcatilli's method, is usually employed to analyze this structure. The *a priori* assumption of Marcatilli's approach is that most of the energy of the modes propagating in the structure is contained within the core region and very little guided power is contained in the corner regions of the guide [34]. Then, the boundary conditions are only matched along the four sides of the hollow region, and the fields in the corners are ignored. The Helmholtz equation in Cartesian coordinates is solved with separated variables, and approximate solutions for the propagation constant and for the field distributions are found. The modes are classified into *E<sup>x</sup> nm* and *<sup>E</sup><sup>y</sup> nm* with most of its electric field polarized in the horizontal or vertical direction, respectively. This approximation gives the mode parameters with sufficient accuracy as long as the assumptions below are valid:

$$\begin{array}{l}(n\lambda/2w)<<1\\(m\lambda/2h)<<1\end{array}\tag{1}$$

$$\begin{array}{l} \left| \overline{\varepsilon} - 1 \right|^{1/2} / \overline{\varepsilon} > > \left( m \lambda / 2w \right) \\\left| \overline{\varepsilon} - 1 \right|^{1/2} / \overline{\varepsilon} > > \left( m \lambda / 2h \right) \end{array} \tag{2}$$

where *n* and *m* are integers describing the *nm*-th mode, *λ* is the free-space wavelength, *w* and *h* the waveguide dimensions, and ¯ is the normalized complex permittivity ¯ = *<sup>r</sup>* − *jσ*/ (*ω*0), where *σ* is the wall conductivity. Finally, a study of modal propagation in curved rectangular tunnels can be found in [20].
