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# 350lect26 - 26 Parallel-plate waveguides TMm modes Last...

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26 Parallel-plate waveguides — TM m modes Last lecture we discussed the TE m modes of propagation in parallel- plate waveguides. z x a σ = σ = These guided modes have y -polarized electric fields transverse to the propagation direction z and exhibit a standing wave pattern in x - direction with m half-wavelengths of variation between the guide plates at x = 0 and x = a . More specifically, the TE m modes have transverse electric field phasors ˜ E = 2 j ˆ yE o e - jk z z sin( k x x ) where k x = m π a , m = 1 , 2 , · · · and k z = k 2 - k 2 x = ω c 1 - f 2 c f 2 with cuto ff frequencies f c = mc 2 a . Alternatively (and equivalently), k z = k 2 - k 2 x = ω c 1 - λ 2 λ 2 c , 1

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with cuto ff wavelengths λ c = 2 a m . Above, the operation frequency f and operation wavelength λ satisfy λ f = c , and furthermore k = 2 π λ = ω c is the operation wavenumber. The propagation characteristics of the guided mode, on the other hand, depends on k z , with v p = ω k z and λ g = 2 π k z denoting the phase velocity and the wavelength of the guided mode when f > f c and, equivalently, λ < λ c , corresponding to propagation condition for a given mode. When f < f c and, equivalently, λ > λ c , the mode is evanescent. Since k z = k 2 - k 2 x = ω c 1 - f 2 c f 2 2
is e ff ectively the dispersion relation of the guided modes having

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