The wavelength scaling of amplitude variance discussed in Section 3.2.4 was found to give good agreement between theory and experiment. We can take this comparison one step further and compare the detailed variations of amplitude at two wavelengths. The amplitude histories for radio-astronomical signals at 81.5 and 118.5 MHz received from a radio source in Cassiopeia [1] are reproduced in Figure 6.1. Investigations of this correlated signal behavior were originally driven by practical considerations. Radio astronomers working with very faint signals wanted to know how wide a frequency band they could use to improve the signal-to-noise ratio. Engineers designing terrestrial and satellite communication relay links wondered whether turbulence scattering would limit the information bandwidth that could be transmitted through the atmosphere.

Both questions are related to the medium bandwidth which measures the frequency separation over which received signals behave in the same way. Severe medium-bandwidth limitations had been observed on tropospheric and ionospheric scatter-propagation circuits. That experience raised the possibility that only a narrow slice of frequency spectrum would be coherent and thus useful for line-of-sight transmissions. Experimental data showed instead that a very large bandwidth should be available [2][3][4][5][6][7]. For weak scattering the signal strength is correlated over a wide range of frequencies because the strong coherent field provides a powerful stabilizing reference – as it cannot for scatter propagation.

History

Quivering of stellar images can be observed with the naked eye and was noted by ancient peoples. Aristotle tried but failed to explain it. A related phenomenon noted by early civilizations was the appearance of shadow bands on white walls just before solar eclipses. When telescopes were introduced, scintillation was observed for stars but not for large planets. Newton correctly identified these effects with atmospheric phenomena and recommended that observatories be located on the highest mountains practicable. Despite these occasional observations, the problem did not receive serious attention until modern times.

How it Began

Electromagnetic scintillation emerged as an important branch of physics following the Second World War. This interest developed primarily in response to the needs of astronomy, communication systems, military applications and atmospheric forecasting. The last fifty years have witnessed a growing, widespread interest in this field, with considerable resources being made available for measurement programs and theoretical research.

Radio signals coming from distant galaxies were detected as this era began, thereby creating the new field of radio astronomy. Microwave receivers developed by the military radar program were used with large apertures to detect these faint signals. Their amplitude varied randomly with time and it was initially suggested that the galactic sources themselves might be changing. Comparison of signals measured at widely separated receivers showed that the scintillation was uncorrelated, indicating that the random modulation was imposed by ionized layers high in the earth's atmosphere.

The electromagnetic field is characterized by the electric and magnetic fields which are vector quantities. The direction taken by the electric-field vector at each point along the path defines the polarization of the field. Faraday rotation of the electric field occurs when a signal propagates through the ionosphere. This rotation is readily observed at microwave frequencies and provides a useful way to measure the integrated electron density along the path.

We are concerned here with the more subtle changes in polarization that are caused by scattering in the lower atmosphere. Significant changes in polarization that are caused by wide-angle scattering in the troposphere are observed for scatter propagation beyond the horizon. By contrast, line-of-sight propagation is dominated by very-small-angle forward scattering. We have assumed so far that the change in polarization for this type of propagation is negligible. That assumption permits one to describe the propagation of light and microwaves in terms of a single scalar quantity. We now need to test this assumption by calculating the depolarization of the incident field.

If the transmitted wave is linearly polarized, we want to estimate how much the electric field rotates as the signal travels through the random medium. Two rather different descriptions of depolarization have been developed and apparently describe different aspects of the same phenomenon. The first is based on diffraction theory.

The second-order term in the Rytov series plays an influential role in determining moments of the field strength and in demonstrating that energy is conserved. The importance of this term was recognized in early studies in which the Rytov approximation was developed to describe propagation in random media [1][2]. Some treatments have used the expected equivalence of the mean irradiance and its free-space values for plane and spherical waves to relate the second-order Rytov term to the phase and amplitude variances estimated to first order [3]. This is often characterized as a consequence of conservation of energy and we will examine that proposition in Chapter 9.

It soon became clear that the preferred approach was to calculate the second-order solution from first principles and then use the result to predict properties of the signal. This was done first for plane waves [1][4][5] and provided reasonable estimates for important properties of signals. That agreement finally established the Rytov approximation as a trustworthy description for the weak-scattering regime [6]. This work was later extended to describe spherical waves [7]. The development of optical lasers stimulated many applications involving beam waves and the corresponding second-order solution was calculated with improving accuracy over the next two decades [8][9][10][11].

The material presented in this chapter is highly analytical and is intended primarily for specialists in the field. The reader who is more interested in applications may wish to proceed directly to Chapters 9 and 10.

The amplitudes of signals passing through the atmosphere change with time in a random manner. The amplitude time history for a microwave signal is reproduced in Figure 5.1 from an early experiment. This rapid variability is due primarily to scattering by irregularities that are being carried through the illuminated volume on prevailing winds. To a lesser extent, it is caused by rearrangement of the irregularities by turbulent mixing and the intrinsic process of subdivision that creates the spectrum, as suggested in Figure 2.5 of Volume 1. Variability of signal amplitude is an important consideration for communication systems and target-location systems. It provides a significant tool for atmospheric scientists and astronomers who exploit it to study the properties of remote regions.

We presented several techniques for describing a moving and evolving turbulent atmosphere in Section 6.1 of Volume 1. The vast majority of propagation studies rely on Taylor's hypothesis to describe the temporal changes that occur in a random medium. This approximation involves two assumptions. It is postulated that the entirety of the turbulent medium is frozen during the measurement interval. It is also assumed that one can ignore the variable component of velocity and that the wind velocity is constant at each location. In combination, these two assumptions imply that the entire air mass is transported at constant speed without being deformed as suggested in Figure 6.2 of Volume 1.

We turn now to the task of describing the variance of amplitude and intensity fluctuations. We avoided this problem in Volume 1 and it is important to recall why we did so. That development was based entirely on solutions for Maxwell's equations generated by the geometrical-optics approximation. The electromagnetic wavelength is completely absent from the eikonal and transport equations that define those solutions. On the other hand, astronomical observations and terrestrial experiments show that the level of scintillation does depend on wavelength. The fundamental problem is that geometrical optics completely ignores diffraction. Yet we know that amplitude fluctuations are caused primarily by small atmospheric irregularities. Diffraction by these small eddies is the principal cause of the amplitude fluctuations that characterize atmospheric scintillation.

Our task in this chapter is to estimate the level of scintillation for optical and microwave propagation. To do so we will depend on the Rytov approximation developed in Chapter 2 that includes diffraction in a rigorous way. It is limited to weak-scattering situations but that covers a broad range of applications. The Rytov approximation gives a complete description for terrestrial microwave links and can often describe millimeter-wave propagation. The same approach characterizes the scintillation experienced by optical and infrared signals that travel near the surface on relatively short paths. By contrast, astronomical observations are characterized by weak scattering unless the source is close to the horizon.

The similarity of signals measured at adjacent receivers is an important feature of electromagnetic scintillation. The way in which their correlation decreases with their separation provides a powerful insight into such phenomena. We discussed the correlation of phase fluctuations measured at nearby receivers in Volume 1. We found there that the phase difference defines the angular accuracy of direction-finding systems and the resolution of interferometric imaging techniques.

In this chapter we will investigate the correlation of signal amplitudes and intensities measured at adjacent receivers. This type of experiment played a crucial role in the development of scintillation physics. Optical measurements of intensity correlation on short paths first validated the description based on Rytov's approximation and Kolmogorov's model of atmospheric turbulence. These experiments were later replicated with microwave and millimeter-wave signals on longer paths.

Diffraction patterns are created on the ground when starlight is scattered by tropospheric irregularities. These patterns are correlated over tens of centimeters and are sometimes visually apparent at the onset of solar eclipses. Astronomical signals are thus spatially correlated over distances that are smaller than most telescope openings. This means that the arriving field is coherent over only a modest portion of the reflector surface. The light-gathering power of astronomical telescopes is limited by this effect unless modern speckle-interferometry techniques are used to reconstruct the original wave-front.

The purpose of this chapter is to lay the foundation for describing phase and amplitude fluctuations on an equal footing. That foundation rests on the Rytov approximation or method of smooth perturbations that was introduced in Chapter 1. This method provides a powerful technique with which one can solve the random-wave equation (1.3) subject to the condition (1.7). For weak scattering of waves in random media, the Rytov approximation provides a significant improvement both on geometrical optics and on the Born approximation.

There are two steps in this approach. The first and most important step is to completely change the structure of the problem. Rytov discovered a transformation of the electric-field strength that changes the random-wave equation to one that can be solved in many interesting situations [1]. The second step is to expand the transformed field strength into a series of terms proportional to successively higher powers of the dielectric variations [2].

The same approach is often applied to the parabolic-wave equation, which represents a special case of (1.3) that describes small-angle forward scattering [3]. We shall use the more general Helmholtz equation here so as to include microwave propagation which often includes wide-angle scattering. We specialize these results using the paraxial approximation for optical and infrared applications – where the scattering angle is usually small. This chapter is necessarily analytical and the reader who is primarily interested in applications may wish to proceed directly to those topics which begin with the next chapter.

Introduction

In this and the following chapter, a study of large circular dipole arrays with one or two elements driven is presented. The top view of such an array is shown in Fig. 11.1a, and a realization of a large circular array with one element driven [1] by monopoles over a ground plane is shown in Fig. 11.1b.

The main reason for initiating the study was the belief that such arrays should possess very narrow resonances if the many parameters of the problem are properly chosen. It was further believed that some particular shapes of non-circular closed-loop arrays might produce a superdirective field pattern. The large circular array of this chapter is the simplest form of the more general closed-loop array. The latter is a subject of ongoing research.

It was seen in Chapter 4 that circular arrays of a small number of elements possess noticeable resonances; previous studies also supported the idea of the existence of very narrow resonances in large circular arrays. For instance, it was known that the long Yagi array may be thought of as a surface-wave structure [2]. Such a structure does not radiate broadside, and it was observed experimentally that this property is preserved if the array is bent into a semi-circle of sufficiently large radius [3].

Essential initial considerations concerning resonances in large circular arrays came from studies in quantum mechanics.

The method of symmetrical components

An array is a configuration of two or more antennas so arranged that the superposition of the electromagnetic fields maintained at distant points by the currents in the individual elements yields a resultant field that fulfils certain desirable directional properties. Since the individual elements in an array are quite close together – the distance between adjacent elements is often a half-wavelength or less – the currents in them necessarily interact. It follows that the distributions of both the amplitude and the phase of the current along each element depend not only on the length, radius, and driving voltage of that element, but also on the distributions in amplitude and phase of the currents along all elements in the array. Since these currents are the primary unknowns from which the radiation field and the driving-point admittance are computed, it is essential that they be determined accurately and not arbitrarily assumed to have identical distributions, as in uniform array theory.

In order to introduce the properties of arrays in a simple and direct manner, it is advantageous to study first the two-element array in some detail. The integral equation (2.15) for the current in a single isolated antenna is readily generalized to apply to the two identical parallel and non-staggered elements shown in Fig. 3.1. It is merely necessary to add to the vector potential on the surface of each element the contributions by the current in the other element.