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Chapter 4:
Free-Space Propagation of Gaussian-Beam Waves
Abstract
Overview: The purpose of this chapter is to introduce the basic features of a Gaussian-beam wave in both the plane of the transmitter and the plane of the receiver. Our main concentration of study involves the lowest-order mode or TEM00 beam, but we also briefly introduce Hermite-Gaussian and Laguerre-Gaussian beams as higher-order modes, or additional solutions, of the paraxial wave equation. Each of these higher-order modes produces a pattern of multiple spots in the receiver plane as opposed to a single (circular) spot from a lowest-order beam wave. Consequently, the analysis of such beams is more complex than that of the TEM00 beam. One advantage in working with the TEM00 Gaussian-beam wave model is that it also includes the limiting classical cases of an infinite plane wave and a spherical wave. We facilitate the free-space analysis of Gaussian-beam waves by introducing two sets of nondimensional beam parameters—one set that characterizes the beam in the plane of the transmitter and another set that does the same in the plane of the receiver. The beam spot radius and phase front radius of curvature, as well as other beam properties, are readily determined from either set of beam parameters. For example, we use the beam parameters to identify the size and location of the beam waist and the geometric focus. The consistent use of these beam parameters in all the remaining chapters of the text facilitates the analysis of Gaussian-beam waves propagating through random media. When optical elements such as aperture stops and lenses exist at various locations along the propagation path, the method of ABCD ray matrices can be used to characterize these elements (including the free-space propagation between elements). By cascading the matrices in sequence, the entire optical path between the input and output planes can be represented by a single 2x—2 matrix. The use of these ray matrices, which is based on the paraxial approximation, greatly simplifies the treatment of propagation through several such optical elements. In later chapters we will extend this technique to propagation paths that also include atmospheric turbulence along portions of the path.
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CHAPTER 4
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