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350lect32 - 32 TM modes in dielectric waveguides Last...

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32 TM modes in dielectric waveguides Last lecture we examined the characteristic equation and the cuto ff frequencies of TE mode of propagation in dielectric slab waveguides. z x d n 2 n 2 n 1 > n 2 θ i Guidance requires θ i > θ c = sin - 1 n 2 n 1 Guided TE m mode fields consisting of the superposition of transverse polarized electric fields ˜ E i = ˆ yE o e - jk 1 ( - cos θ i x +sin θ i z ) and ˜ E r = ˆ yE o Γ TE e - jk 1 (cos θ i x +sin θ i z ) , where Γ TE = n 1 cos θ i + j n 2 1 s in 2 θ i - n 2 2 n 1 cos θ i - j n 2 1 s in 2 θ i - n 2 2 , have 1. propagation angles θ i > θ c = s in - 1 n 2 n 1 , (critical angle) 2. satisfying a characteristic equation d v 1 /f cos θ i - m 2 = 1 π ta n - 1 s in 2 θ i - n 2 2 / n 2 1 cos θ i , m = 0 , 1 , 2 , 3 , · · · 3. for frequencies f exceeding the cuto ff frequency f c = mc 2 d n 2 1 - n 2 2 , m = 0 , 1 , 2 , 3 · · · 1
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For a given f = 2 π ω , the charcteristic equation can be solved (typically by using graphical techniques) for θ i , from which we can calculate the propagation constant k z = k 1 s in
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