Introduction

We have stated a⃗ this in Chapter 7 and we state it again: Maxwell's equations contain all the information necessary to characterize the electromagnetic fields at any point in a medium. For the electromagnetic (EM) fields to exist they must satisfy the four Maxwell equations at the source where they are generated, at any point in a medium through which they propagate, and at the load where they are received or absorbed.

In this chapter, we concentrate mainly on the propagation of EM fields in a source-free medium. As the fields must satisfy the four coupled Maxwell equations involving four unknown variables, we first obtain an equation in terms of one unknown variable. Similar equations can then be obtained for the other variables. We refer to these equations as the general wave equations. We will show in Chapter 11 that the fields generated by time-varying sources propagate as spherical waves. However, in a small region far away from the radiating source, the spherical wave may be approximated as a plane wave, that is, one in which all the field quantities are in a plane normal to the direction of its propagation (the transverse plane). Consequently, a plane wave does not have any field component in its direction of propagation (the longitudinal direction).

We first seek the solution of a plane wave in an unbounded dielectric medium and show that the wave travels with the speed of light in free space.

Introduction

In our discussion of transmission lines, we pointed out that the resistance of a conductor increases with an increase in the signal frequency, leading to an increase in power loss along the line. This power loss becomes intolerable at microwave frequencies (in the GHz range) and makes the transmission line almost impractical. At such high frequencies hollow conductors, known as waveguides, are employed to guide electrical signals efficiently. Figure 10.1 shows a typical waveguide assembly.

In the study of transmission lines with at least two conductors, we found that the propagating wave has field components in the transverse direction and is referred to as the transverse electromagnetic (TEM) wave. However, as a waveguide consists of only one hollow conductor, we do not expect it to support the TEM wave. In this chapter, we show that a waveguide can support the other two types of waves, the transverse magnetic (TM) and the transverse electric (TE) waves. These waves can exist inside a hollow conductor under certain conditions. TM and TE waves can also propagate in a region bounded by a parallel-plate transmission line, in which case the two conducting plates are said to form a parallel-plate waveguide.

The propagation of an electromagnetic wave inside a waveguide is quite different than the propagation of a TEM wave. When a wave is introduced at one end of the waveguide, it is reflected from the wall of the waveguide whenever it strikes it.

Predicting the distribution of measured fluctuations in intensity is one of the great challenges in describing electromagnetic propagation through random media. The amplitude variance addressed in Chapter 3 describes the width of this distribution but tells one nothing about the likelihood that very large or very small fluctuations will occur. By contrast, the probability density function for intensity variations provides a complete portrait of the signal's behavior. This wider perspective is important for engineering applications in which one must predict the complete range of signal values. The same description provides an important insight into the physics of scattering of waves by turbulent irregularities.

The Rytov approximation predicts that the distribution is log-normal. That forecast is confirmed by measurements made over a wide range of conditions on terrestrial links. It is also confirmed by astronomical observations. The agreement is independent of the model of turbulent irregularities employed. It therefore provides a test for the basic theoretical approach to electromagnetic propagation. Second-order refinements to the log-normal distribution predict a skewed log-normal distribution and agree with numerical simulations. This success represents a significant achievement for the Rytov approach.

The distribution provided by Rytov theory needs to be enlarged in some situations. It describes the short-term fluctuations observed over periods of a few minutes to a few hours. One must also consider variations of signal strength measured over weeks, months or years. Long-term statistics are important for communication services employing terrestrial or satellite relays.

All of the development so far has concentrated on the variances of phase and amplitude – or their correlations with respect to separation, time delay and frequency separation. We turn now to moments of the field strength itself. These are the quantities that are often measured in astronomical observations and terrestrial experiments. They are usually calculated with theories that characterize strong scintillation. It is significant that the Rytov approximation gives identical results for many of these quantities. This approach requires only the tools we have already developed.

The average field strength and mean irradiance set important reference levels for measurement programs. They are also needed in order to describe other features of propagation in random media. Accurate estimates for these quantities require both first- and second-order solutions in the Rytov expansion. These depend on the double-scattering expressions which were established for plane, spherical and beam waves in Chapter 8. Field-strength moments are more difficult to estimate because they contain the Born terms in exponential form. We shall learn that the mean field is attenuated very rapidly with distance. By contrast, the mean irradiance for plane and spherical waves is everywhere equal to its free-space value.

These calculations set the stage for analyzing two important features of the electromagnetic field. With the second Rytov approximation one can demonstrate that energy is conserved in a nonabsorbing medium. The mutual coherence function is calculated in the same way and provides an expression identical to that predicted both by geometrical optics and by strong-scattering theories.

The first volume on electromagnetic scintillation exploited geometrical optics to describe propagation in random media. That method represents an approximate solution for Maxwell's equations, which define the electromagnetic field. It was surprisingly successful in two important respects, even though it completely ignores diffraction effects.

Geometrical optics provides an accurate description for the signal-phase fluctuations imposed by a random medium. In this approximation, phase and range variations are caused by random speeding up and slowing down of the signal as it travels along the nominal ray trajectory. The phase variance estimated in this way is proportional to the distance traveled and to the first moment of the spectrum of refractive irregularities. It is therefore primarily sensitive to large eddies and diffraction effects can be safely ignored. This description is confirmed over an unusually wide range of wavelengths and propagation conditions.

The same technique was used to describe the phase difference measured between adjacent receivers. That result is needed in order to interpret observations made with microwave and optical interferometers. A similar expression characterizes the angular resolution of large telescopes in the limit of small separations. In this approach, angular errors are caused by random refractive bending of the rays as they travel through the random medium. The predicted resolution is proportional to the distance traveled and to the spectrum's third moment.