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Wave Guide

Parallel Plate Waveguide¶

TE Waves¶

\begin{matrix} \vec{E} = E_{0} \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{y}} \\ \vec{H} = - \frac{E_{0}}{η} \sin θ \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{x}} + \frac{E_{0}}{η} \cos θ \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{z}} \end{matrix}

cutoff wavelength

\begin{matrix} a = m \frac{λ_{x}}{2} = \frac{m \frac{λ}{\cos θ}}{2} \\ ⟹ λ = \frac{2 a}{m} \cos θ \\ ⟹ λ_{c} = [λ]_{\cos θ = 1} = \frac{2 a}{m} \\ ⟹ \cos θ = \frac{λ}{λ_{c}} \end{matrix}

guide wavelength (in $z$ direction)

\begin{matrix} λ_{g} = \frac{2 π}{k_{z}} = \frac{2 π}{k \sin θ} = \frac{λ}{\sin θ} \\ ⟹ \sin θ = \frac{λ}{λ_{g}} \end{matrix}

by the $λ_{c}$ and $λ_{g}$ we can also write the equations as

\begin{matrix} \vec{E} = E_{0} \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{y}} \\ \vec{H} = - \frac{E_{0}}{η} \frac{λ}{λ_{g}} \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{x}} + \frac{E_{0}}{η} \frac{λ}{λ_{c}} \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{z}} \end{matrix}

TM Waves¶

\begin{matrix} \vec{H} = H_{0} \cos (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{y}} \end{matrix}

\begin{aligned} \vec{E} & = η H_{0} \sin θ \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{x}} + η H_{0} \cos θ \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{z}} \\ = η H_{0} \frac{λ}{λ_{g}} \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{x}} + η H_{0} \frac{λ}{λ_{c}} \sin (k_{x} x) \sin (ω t - k_{z} z) \hat{a_{z}} \end{aligned}

Rectangular Waveguide¶

$γ$

\begin{matrix} k^{2} = k_{x}^{2} + k_{y}^{2} + k_{z}^{2} = k_{x}^{2} + k_{y}^{2} + (- j γ)^{2} \\ ⟹ γ^{2} = k_{t}^{2} - k^{2} \end{matrix}

TM mode¶

tip

notice that we assuming perfect conductors.

thus,

\begin{matrix} {\vec{E}}_{n} = 0 \\ ⟹ E_{z s} = 0, at boundary \end{matrix}

\begin{matrix} E_{z s} = E_{0} \sin (\frac{m π x}{a}) \sin (\frac{n π y}{b}) e^{- γ z} \end{matrix}

TE mode¶

\begin{matrix} H_{z s} = H_{0} \cos (\frac{m π x}{a}) \cos (\frac{n π y}{b}) e^{- γ z} \end{matrix}

Resonators¶

TM

\begin{aligned} E_{z s} & = \frac{E_{0}}{2} \sin (\frac{m π x}{a}) \sin (\frac{n π y}{b}) (e^{- j k_{z} z} + e^{j k_{z} z}) \\ = E_{0} \sin (\frac{m π x}{a}) \sin (\frac{n π y}{b}) \cos (\frac{p π z}{d}) \end{aligned}

TE

\begin{aligned} H_{z s} & = H_{0} \cos (\frac{m π x}{a}) \cos (\frac{n π y}{b}) \sin (\frac{p π z}{d}) \end{aligned}

Q Factors¶

\begin{matrix} Q = 2 π \frac{W}{P_{L} T} = \frac{ω W}{P_{L}} \end{matrix}

recall

\begin{matrix} Q = \frac{X}{R} = \frac{ω L}{R} = \frac{ω I^{2} L}{I^{2} R} = \frac{ω W}{P} \end{matrix}

Dominant Mode¶

the mode with lowest $f_{c}$ (i.e. largest $λ_{c}$ ).

Cylindrical¶

$β_{c}$

\begin{matrix} k^{2} = β_{c}^{2} + β^{2} \end{matrix}

TM mode¶

\begin{matrix} E_{z s} = J_{n} [β_{c} ρ] (A_{n} \cos n ϕ + B_{n} \sin n ϕ) e^{\mp j β z} \end{matrix}

TE mode¶

\begin{matrix} H_{z s} = J_{n} [β_{c} ρ] (A_{n} \cos n ϕ + B_{n} \sin n ϕ) e^{\mp j β z} \end{matrix}

Dielectric Slab Waveguide¶

cutoff wavelength

\begin{matrix} λ_{c} = \frac{2 d \sqrt{ε_{r 1} - ε_{r 2}}}{m} m = 0, 1, 2, \dots \end{matrix}

thinking

\begin{matrix} k \cdot 2 d \cdot \cos θ_{i} - 2 ∠ Γ = 2 m π \\ ⟹ \frac{2 π \sqrt{ε_{r 1}}}{λ_{0}} \cdot d \cdot \cos θ_{i} - ∠ Γ = m π \\ ⟹ ∠ Γ = k \cdot d \cdot \cos θ_{i} - m π \end{matrix}

when cutoff $∠ Γ = 0$

\begin{matrix} k_{c} d \cos θ_{i} = m π \\ ⟹ \frac{2 π \sqrt{ε_{r 1}}}{λ_{c}} d \cos θ_{i} = m π \\ \frac{2 π \sqrt{ε_{r 1}}}{λ_{c}} d \sqrt{1 - \frac{ε_{2}}{ε_{1}}} = m π \\ ⟹ λ_{c} = \frac{2 d \sqrt{ε_{r 1} - ε_{r 2}}}{m} \end{matrix}