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Unformatted text preview: Waveguides and Cavities John William Strutt also known as Lord Rayleigh (1842  1919) September 17, 2001 Contents 1 Reflection and Transmission at a Conducting Wall 2 1.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Power and Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Wave Guides 7 2.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Transverse Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 TEM Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 TE and TM Modes . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Energy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.1 TE Modes in Rectangular and Circular Guides . . . . . . . . . 15 3 Attenuation of Modes in Waveguides 16 1 4 Resonating cavities 18 2 In this chapter we continue with the topic of solutions of the Maxwell equations in the form of waves. This time we seek solutions in the presence of bounding surfaces which may take a variety of forms. The basic possibilities are to have boundaries in 1. one dimension only, such as a pair of parallel planes; 2. two dimensions, such as several intersecting planes forming a pipe or channel; and 3. three dimensions, such as a collection of intersecting planes that completely bound some region of space. The materials employed to form the boundaries are usually 1 conductors. The mathe matical problem is a boundaryvalue problem for solutions of the Maxwell equations. We shall look at harmonic solutions within the cavity or channel and must match these solutions onto appropriate ones within the walls or bounding materials. If the walls are constructed from a good conductor, the boundary conditions become sim ple and the boundaryvalue problem itself is not too difficult. This point is explored in the following sections. 1 Reflection and Transmission at a Conducting Wall We consider the reflection and transmission of a harmonic plane wave incident on a conducting material at a planar surface. We let the incident wave have an arbitrary angle of incidence  which gives a hard problem to solve in the general case  and then imagine that the conductivity is very large  which simplifies the solution by allowing an expansion in a small parameter. Physically, the central point is that if &gt;&gt; , 1 Dielectric materials are also used, with conditions such that total internal reflection takes place at the surfaces in order to keep the wave within the channel or cavity. 3 then the skin depth = c/ 2 of the wave in the conductor is much smaller than the wavelength of the incident wave. The distance over which the fields vary in the conductor depends on the direction. In the direction normal to the surface, this distance is ; in directions parallel to the surface, it is . Thus by having &lt;&lt; , we can often ignore variations of the fields parallel to the surface in comparison with...
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 Fall '11
 Electrodynamics
 Power

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