Question 13.7.Q1: Electromagnetic (EM) waveguides are used for transmission of......

(a) The basic principles behind transmission EM waveguides and acceleration EM waveguides are the same; however, there are several notable differences between the two types of waveguide with respect to: (1) Design, (2) Cross section, (3) Operating mode, (4) Core material, and (5) Microwave phase velocity. Summary of notable differences is provided in Table 13.6.

(1) Design. Transmission waveguides are uniform in cross section meaning that their cross section does not change along the direction of RF propagation. Acceleration waveguides, on the other hand, are non-uniform meaning that their cross section varies periodically along the direction of RF propagation. They are loaded with disks that define distinct cavities in the acceleration waveguide and cause partial reflection of the RF wave in order to slow down the phase velocity below the speed of light in vacuum.

(2) Cross section. Transmission waveguide most often has a rectangular cross section with sides a and b where a>b, while the cross section of accelerator waveguide is circular with basic core radius a and disk radius b where a>b. In addition to rectangular transmission waveguides, it is possible to have circular transmission EM waveguides; however, all acceleration waveguides are circular.

(3) Operating mode. Transmission waveguides are usually designed such that only the lowest (dominant) transverse microwave mode can propagate through the waveguide. This is the transverse electric TE_{11} mode characterized by m = 1 and n = 1. Particle acceleration, on the other hand, is carried out with the lowest (dominant) transverse magnetic \left(TM_{01}\right) mode characterized by m = 0 and n = 1, since this is the lowest special mode with E_z oriented in the direction of particle motion; a necessary condition for charged particle acceleration.

(4) Core medium. Transmission waveguides are usually filled with a pressurized dielectric gas, however, it is also possible to transmit microwaves in evacuated transmission waveguides; acceleration waveguides are always evacuated.

(5) Phase velocity. In transmission waveguides the phase velocity υ_{ph} of the RF wave exceeds the speed of light c in vacuum (υ_{ph} > c); acceleration waveguides, on the other hand, are designed such that υ_{ph} is slowed down to slightly below c in order to allow the charged particle to follow the RF wave.

(b) Electric field E and magnetic field B in the core of a uniform circular evacuated EM waveguide are vectors with three components, each component depending on three spatial coordinates and one temporal coordinate

\begin{aligned} {E} & =\left[{E}_r(r, \theta, z, t), {E}_\theta(r, \theta, z, t), {E}_z(r, \theta, z, t)\right] \text { and } \\ \mathcal{B} & =\left[ {\mathcal{B}_r}(r, \theta, z, t), \mathcal{B}_r(r, \theta, z, t), \mathcal{B}_r(r, \theta, z, t)\right] \end{aligned}

(1) The E_z\ and\ B_z components of E and B for a uniform evacuated circular EM waveguide are given by the following expressions

and

where γ_n is a parameter determined from boundary conditions and related to free space wave number k and waveguide wave number (waveguide propagation coefficient) k_{\mathrm{g}} \text { as } \gamma_n^2=k^2-k_{\mathrm{g}}^2, \varphi=k_{\mathrm{g}} z-\omega t is the phase of the RF wave, and A_{m n}, B_{m n} \text {, }\ C_{mn},\ and\ D_{mn} are coefficients that are determined from initial conditions.

(2) Components E_z\ and\ B_z are determined from wave equations for E_z\ and\ B_z.

\nabla^2 {E}_z=\frac{1}{c^2} \frac{\partial^2 {E}_z}{\partial^2 t^2}          (13.203)

and

\nabla^2 \mathcal{B}_z=\frac{1}{c^2} \frac{\partial^2 \mathcal{B}_z}{\partial^2 t^2}          (13.204)

where c is the speed of light in vacuum and ∇² is the scalar Laplacian operator expressed in cylindrical coordinates for circular EM waveguide as follows

\nabla^2=\frac{\partial^2}{\partial r^2}+\frac{1}{r} \frac{\partial}{\partial r}+\frac{1}{r^2} \frac{\partial^2}{\partial \theta^2}+\frac{\partial^2}{\partial z^2} .           (13.205)

Wave equations (13.203) and (13.204) are linear partial differential equations of the second order in four variables (3 spatial and one temporal) with constant coefficients. The most common method for solving the two wave equations is the method of separation of variables leading to solutions for the z components E_z\ and\ B_z of the electric field E and magnetic field B in the core of the waveguide.

(3) Once E_z\ and\ B_z are known, the other components of E and B in an evacuated circular EM waveguide are determined from Maxwell equations for free space expressed as follows

\nabla \cdot {E}=0         (13.206)

\nabla \cdot \boldsymbol{B}=0,           (13.207)

\nabla \times {E}=-\frac{\partial \mathcal{B}}{\partial t}         (13.208)

\nabla \times \mathcal{B}=\frac{1}{c^2} \frac{\partial {E}}{\partial t}          (13.209)

with (∇·) the divergence and (∇×) the curl on vectors E and B.

(c) We start the derivation of components E_r,\ E_θ ,\ B_r,\ and\ B_θ for circular EM waveguide with Maxwell equations (13.206) and (13.207) as ∇ ·E = 0 and ∇ ·B = 0, respectively, and express them in cylindrical coordinates as follows

\nabla \cdot {E}=\frac{1}{r} \frac{\partial}{\partial r}\left(r {E}_r\right)+\frac{1}{r} \frac{\partial {E}_\theta}{\partial \theta}+\frac{\partial {E}_z}{\partial z}=\frac{1}{r} \frac{\partial}{\partial r}\left(r {E}_r\right)+\frac{1}{r} \frac{\partial {E}_\theta}{\partial \theta}+i k_z E_z=0        (13.210)

and

\nabla \cdot \boldsymbol{B}=\frac{1}{r} \frac{\partial}{\partial r}\left(r \mathcal{B}_r\right)+\frac{1}{r} \frac{\partial \mathcal{B}_\theta}{\partial \theta}+\frac{\partial \mathcal{B}_z}{\partial z}=\frac{1}{r} \frac{\partial}{\partial r}\left(r \mathcal{B}_r\right)+\frac{1}{r} \frac{\partial \mathcal{B}_\theta}{\partial \theta}+i k_z B_z=0 .             (13.211)

Next, we express (13.208) and (13.209), respectively, in cylindrical coordinates as

and

where \hat{\mathbf{r}}, \hat{\boldsymbol{\Theta}} \text {, and } \hat{\mathbf{z}} are the standard unit vectors in the cylindrical coordinate system. Equations (13.212) and (13.213) have the following components of the curl operator [Note: (13.214), (13.215), and (13.216) follow from (13.212); (13.217), (13.218), and (13.219) from (13.213)]

\frac{1}{r} \frac{\partial {E}_z}{\partial \theta}-\frac{\partial {E}_\theta}{\partial z}=i \omega \mathcal{B}_r=\frac{1}{r} \frac{\partial {E}_z}{\partial \theta}-i k_z {E}_\theta \quad \text { or } \quad \mathcal{B}_r=-\frac{i}{\omega r} \frac{\partial {E}_z}{\partial \theta}-\frac{k_{\mathrm{g}}}{\omega} {E}_\theta,          (13.214)

\frac{\partial {E}_z}{\partial z}-\frac{\partial {E}_z}{\partial r}=i \omega \mathcal{B}_\theta=i k_{\mathrm{g}} {E}_r-\frac{\partial {E}_z}{\partial r} \quad \text { or } \quad \mathcal{B}_\theta=\frac{k_{\mathrm{g}}}{\omega} {E}_r+\frac{i}{\omega} \frac{\partial {E}_z}{\partial r},        (13.215)

{E}_r=\frac{i c^2}{\omega r} \frac{\partial \mathcal{B}_z}{\partial \theta}+\frac{k_{\mathrm{g}} c^2}{\omega} \mathcal{B}_\theta,          (13.217)

\frac{\partial \mathcal{B}_r}{\partial z}-\frac{\partial \mathcal{B}_z}{\partial r}=-\frac{i \omega}{c^2} {E}_\theta=i k_{\mathrm{g}} \mathcal{B}_r-\frac{\partial \mathcal{B}_z}{\partial r} \quad \text { or } \quad {E}_\theta=-\frac{k_{\mathrm{g}} c^2}{\omega} \mathcal{B}_r-\frac{i c^2}{\omega} \frac{\partial \mathcal{B}_z}{\partial r}          (13.218)

\frac{1}{r} \frac{\partial\left(r \mathcal{B}_\theta\right)}{\partial r}-\frac{\partial \mathcal{B}_r}{\partial \theta}=-\frac{i \omega}{c^2} {E}_z \quad \text { or } \quad {E}_z=\frac{i c^2}{\omega}\left[\frac{\partial \mathcal{B}_y}{\partial x}-\frac{\partial \mathcal{B}_x}{\partial y}\right]           (13.219)

Pairing up appropriate equations in the group from (13.214) to (13.219), we can now determine components E_r,\ E_θ ,\ B_r,\ and\ B_θ as follows:

(1) Inserting B_θ of (13.215) into (13.217) gives the following expression for component E_r.

{E}_r=i\left[\frac{k_{\mathrm{g}} c^2}{\omega^2} \frac{\partial {E}_z}{\partial r}+\frac{c^2}{\omega r} \frac{\partial \mathcal{B}_z}{\partial \theta}\right]\left(1-\frac{k_{\mathrm{g}}^2 c^2}{\omega^2}\right)^{-1}=\frac{i}{\gamma_n^2}\left[k_{\mathrm{g}} \frac{\partial {E}_z}{\partial r}+\frac{\omega}{r} \frac{\partial \mathcal{B}_z}{\partial \theta}\right]         (13.220)

(2) Inserting B_r of (13.214) into (13.218) gives the following expression for component E_θ.

{E}_\theta=i\left[-\frac{c^2}{\omega} \frac{\partial \mathcal{B}_z}{\partial r}+\frac{k_{\mathrm{g}} c^2}{\omega^2 r} \frac{\partial {E}_z}{\partial \theta}\right]\left(1-\frac{k_{\mathrm{g}}^2 c^2}{\omega^2}\right)^{-1}=\frac{i}{\gamma_n^2}\left[-\omega \frac{\partial \mathcal{B}_z}{\partial r}+\frac{k_{\mathrm{g}}}{r} \frac{\partial {E}_z}{\partial \theta}\right] .           (13.221)

(3) Inserting E_θ of (13.218) into (13.214) gives the following expression for component B_r.

\mathcal{B}_r=i\left[\frac{k_{\mathrm{g}} c^2}{\omega^2} \frac{\partial \mathcal{B}_z}{\partial r}-\frac{1}{\omega r} \frac{\partial {E}_z}{\partial \theta}\right]\left(1-\frac{k_{\mathrm{g}}^2 c^2}{\omega^2}\right)^{-1}=\frac{i}{\gamma_n^2}\left[k_{\mathrm{g}} \frac{\partial \mathcal{B}_z}{\partial r}+\frac{\omega}{c^2 r} \frac{\partial {E}_z}{\partial \theta}\right]          (13.222)

(4) Inserting E_r of (13.217) into (13.215) gives the following expression for component B_θ.

\mathcal{B}_\theta=i\left[\frac{1}{\omega} \frac{\partial {E}_z}{\partial r}+\frac{k_{\mathrm{g}} c^2}{\omega^2 r} \frac{\partial \mathcal{B}_z}{\partial \theta}\right]\left(1-\frac{k_{\mathrm{g}}^2 c^2}{\omega^2}\right)^{-1}=\frac{i}{\gamma_n^2}\left[\frac{\omega}{c^2} \frac{\partial {E}_z}{\partial r}+\frac{k_{\mathrm{g}}}{r} \frac{\partial \mathcal{B}_z}{\partial \theta}\right]          (13.223)

where \gamma_n^2=k^2-k_{\mathrm{g}}^2 \text { and } \gamma=k c \text { where } k is the free space propagation coefficient. Equations (TT) through (13.223) show that the transverse components E_r,\ E_θ ,\ B_r,\ and\ B_θ can be determined with relative ease using Maxwell equations for free space in conjunction with known axial components E_z and B_z that are determined from appropriate wave equations (13.203) and (13.204), respectively.

(d) Equations (13.220) through (13.223) give general expressions for transverse components E_r,\ E_θ ,\ B_r,\ and\ B_θ as a function of axial components E_z\ and\ B_z for a uniform circular EM waveguide. We now determine the transverse components for the three special modes: transverse magnetic (TM), transverse electric (TE), and transverse electromagnetic (TEM) that are characterized as follows:

(1) TM modes: B_z = 0 everywhere inside the waveguide core and the Dirichlettype boundary condition \left. {E}_z\right|_{r=a}=0 \text { applies to } {E}_z at the boundary between the waveguide core and waveguide wall resulting in the following expression for γ_n

\gamma_n=\frac{x_{m n}}{a},           (13.224)

where x_{mn} is the n-th zero of the m-th order Bessel function.
The transverse components E_r,\ E_θ ,\ B_r,\ and\ B_θ are now from (13.220) through (13.223) simplified as follows, recognizing that ∂B_z/∂r = ∂B_z/∂θ = 0, since for TM modes B_z = 0 everywhere inside the waveguide core

{E}_r=\frac{i k_{\mathrm{g}}}{\gamma_n^2} \frac{\partial {E}_z}{\partial r},         (13.225)

{E}_\theta=\frac{i k_{\mathrm{g}}}{\gamma_n^2 r} \frac{\partial {E}_z}{\partial \theta}           (13.226)

\mathcal{B}_r=\frac{i \omega}{\gamma_n^2 c^2 r} \frac{\partial {E}_z}{\partial \theta},            (13.227)

\mathcal{B}_\theta=\frac{i \omega}{\gamma_n^2 c^2} \frac{\partial {E}_z}{\partial r}           (13.228)

(2) TE modes: E_z = 0 everywhere inside the waveguide core and the Neumanntype boundary condition \partial \mathcal{B}_z /\left.\partial r\right|_{r=a}=0 \text { applies to } \partial \mathcal{B}_z / \partial r at the boundary between the waveguide core and waveguide wall resulting in the following expression for γ_n.

\gamma_n=\frac{y_{m n}}{a}           (13.229)

where y_{mn} is the n-th zero of the first derivative of the m-th order Bessel function.
The transverse components E_r,\ E_θ ,\ B_r,\ and\ B_θ are now from (13.220) through (13.223) given as follows, recognizing that ∂E_z/∂r = ∂E_z/∂θ = 0,\ since\ E_z = 0 everywhere in the waveguide core

{E}_r=\frac{i \omega}{\gamma_n^2 r} \frac{\partial \mathcal{B}_z}{\partial \theta},          (13.230)

{E}_\theta=-\frac{i \omega}{\gamma_n^2} \frac{\partial \mathcal{B}_z}{\partial r},           (13.231)

\mathcal{B}_r=\frac{i k_{\mathrm{g}}}{\gamma_n^2} \frac{\partial \mathcal{B}_z}{\partial r}           (13.232)

\mathcal{B}_\theta=\frac{i k_{\mathrm{g}}}{\gamma_n^2 r} \frac{\partial \mathcal{B}_z}{\partial \theta} .          (13.233)

(3) TEM mode: Both E_z = 0\ and\ B_z = 0 everywhere and (13.220) through (13.223) show that all transverse components E_r,\ E_θ ,\ B_r,\ and\ B_θ are also equal to zero. We conclude that TEM modes cannot propagate through uniform circular EM waveguides.

(e) Components of lowest (dominant) TM and TE modes in a uniform evacuated circular EM waveguide with radius a are determined using the following steps:
The general expressions for z components E_z\ and\ B_z of electric field E and magnetic field B, respectively, given in (13.201) and (13.202), respectively, are used here to determine E_z\ and\ B_z for the dominant TM and TE modes, respectively. Expressions for the other four components E_r,\ E_θ ,\ B_r,\ and\ B_θ for the special modes were derived in (d). We now use these expressions to determine the electric and magnetic field components for the dominant TM and TE modes.

(1) Transverse magnetic (TM) modes are characterized by B_z = 0 everywhere in the waveguide core and the dominant (lowest) TM mode occurs for m = 0 and n = 1, resulting in the following expression for parameter γ_n from (13.224)

\gamma_n=\frac{x_{01}}{a}=\frac{2.405}{a}           (13.234)

(i) The electric field component E_z for the dominant TM_{01} mode is now from (13.201) given as

{E}_z={E}_{01} J_0\left(\gamma_1 r\right) e^{i \varphi}={E}_{01} J_0\left(\frac{x_{01}}{a} r\right) e^{i \varphi}={E}_{01} J_0\left(\frac{2.405}{a} r\right) e^{i \varphi}            (13.235)

(ii) The magnetic field component B_z = 0 everywhere in the waveguide core for TM modes.
(iii) The electric field component E_r is determined from (aa) using B_z = 0,\ E_z from (13.235), and γ_n from (13.234) for m = 0 and n = 1 to get the following result

where we used x_{01} = 2.405 and the following recursive relationship for Bessel function J_m(x) \frac{\mathrm{d} J_m(x)}{\mathrm{d} x}=-J_{m+1}(x)+\frac{m}{x} J_m(x) \text { resulting in } \frac{\mathrm{d} J_0(x)}{\mathrm{d} x}= -J_1(x) \text { or }-\frac{\mathrm{d}}{\mathrm{dr}}  J_0\left(\frac{x_{01}}{a} r\right)=-\frac{x_{01}}{a} J_1\left(\frac{x_{01}}{a} r\right) .

(iv) The electric field component E_θ of (13.226) is zero because ∂E_z/∂θ = 0.

{E}_\theta=i \frac{k_{\mathrm{g}}}{\gamma_n^2 r} \frac{\partial {E}_z}{\partial \theta}=0 .         (13.237)

(v) The magnetic field component B_r of (13.227) is zero because ∂E_z/∂θ = 0.

\mathcal{B}_r=i \frac{\omega}{\gamma_n^2 c^2} \frac{\partial {E}_z}{\partial \theta}=0 .             (13.238)

(vi) The magnetic field component B_θ for the dominant TM_{01} mode is determined from (13.228) using B_z = 0,\ E_z from (13.235), and γ_n from (13.234) for m = 0 and n = 1 as follows

where we used again \frac{\mathrm{d}}{\mathrm{d} r} J_0\left(\frac{x_{01}}{a} r\right)=-\frac{x_{01}}{a} J_1\left(\frac{x_{01}}{a} r\right) .

(2) Transverse electric (TE) modes are characterized by E_z = 0 everywhere in the waveguide core and the dominant (lowest) TE mode occurs for m = 1 and n = 1,

resulting in the following expression for parameter γ_n from (13.229)

\gamma_n=\frac{y_{11}}{a}=\frac{1.841}{a} .         (13.240)

(i) The magnetic field component B_z for the dominant TE_{11} mode is from (13.202) given as

\mathcal{B}_z=\mathcal{B}_{11} J_0\left(\gamma_1 r\right) e^{i \varphi}=\mathcal{B}_{11} J_0\left(\frac{y_{11}}{a} r\right) e^{i \varphi}=\mathcal{B}_{11} J_0\left(\frac{1.841}{a} r\right) e^{i \varphi} .           (13.241)

(ii) The electric field component E_z = 0 everywhere in the waveguide core for TE modes.
(iii) The electric field component E_r of (13.230) is zero because ∂B_z/∂θ = 0.

{E}_r=i \frac{\omega}{\gamma_n^2 r} \frac{\partial \mathcal{B}_z}{\partial \theta}=0 .           (13.242)

(iv) The electric field component E_θ for the dominant TE_{11} mode is determined from (13.231) using E_z = 0,\ B_z from (13.241), and γ_n from (13.240) for m = 1 and n = 1 as follows

where we used y_{11} = 1.841 and the following recursive relationship for Bessel function J_m(x).

\frac{\mathrm{d} J_m(x)}{\mathrm{d} x}=-J_{m+1}(x)+\frac{m}{x} J_m(x) that results in \frac{\mathrm{d} J_0(x)}{\mathrm{dx}}=-J_1(x)\quad or \\   \frac{\mathrm{d}}{\mathrm{d} r} J_0\left(\frac{y_{11}}{a} r\right)=-\frac{y_{11}}{a} J_1\left(\frac{y_{11}}{a} r\right)

(v) The magnetic field component B_r for the dominant TE_{11} mode is determined from (13.232) using E_z = 0,\ B_z from (13.241), and γ_n from (13.240) for m = 1 and n = 1 as follows

where we used again \frac{\mathrm{d}}{\mathrm{d} r} J_0\left(\frac{y_{11}}{a} r\right)=-\frac{y_{11}}{a} J_1\left(\frac{y_{11}}{a} r\right) .

(vi) The magnetic field component B_θ of (13.233) is zero because ∂B_z/∂θ =0.

\mathcal{B}_\theta=i \frac{k_{\mathrm{g}}}{\gamma_n^2 r} \frac{\partial \mathcal{B}_z}{\partial \theta}=0            (13.245)