Question 13.3.Q1: Electromagnetic (EM) waveguide is a device used for propagat......

(a) Electromagnetic (EM) uniform waveguide is a hollow metallic pipe used for transmission of microwaves from the source of microwave power (magnetron or klystron) to user (transmitting antenna in radar installations, cooking chamber in microwave ovens, accelerating waveguide in particle linacs, etc.). The main characteristics of a uniform waveguide are:

(1) The cross sectional profile of a uniform waveguide is most commonly rectangular (with longer side a and shorter side b) or circular (with radius a) and, as the name implies, uniform (i.e., the cross sectional profile is smooth and does not change along the axis of the waveguide). Figure 13.3 shows the geometry used for rectangular waveguides in (A) and circular waveguides in (B). It is assumed that the axis of the waveguide tube is oriented in the direction of the z-axis of the Cartesian coordinate system for rectangular waveguides and cylindrical coordinate system for circular waveguides.
(2) Walls of uniform waveguides are made of a conducting medium, such as copper; core of the uniform waveguide is either evacuated or more commonly filled with pressurized (∼2 atm) dielectric gas, such as sulfur hexafluoride \left(SF_6\right).
(3) The cross sectional dimensions of a typical uniform EM waveguide are of the order of the wavelength of the RF waves that the waveguide transmits.
(4) Radiofrequency waves propagate in a uniform waveguide with phase velocity υ_{ph} that exceeds the speed of light c in vacuum and with group velocity υ_{gr} that is between 0 and c. Since υ_{gr} is generally less than c, uniform waveguides can be used for transmission of radiofrequency power but cannot be used for acceleration of charged particles in linear accelerators (linacs).
(5) EM waveguides function as a high pass filter. This means that to propagate in a given waveguide the RF frequency must exceed a certain minimum frequency referred to as the cutoff frequency of the waveguide. Waveguides also function as wideband devices and are used for transmission of RF power or communication signals.

(b) Waveguides are analyzed by solving the wave equations for the electric field {E} and magnetic field \mathcal{B} in the core of the waveguide in conjunction with boundary conditions that account for waveguide wall and core materials as well as waveguide geometry. The wave equations are partial differential equations of the second order derived from Maxwell equations. They have multiple solutions or modes, each mode categorized by its minimum frequency, called cutoff frequency that can be transmitted through the waveguide.
The general boundary conditions for EM waveguides with perfect copper conductor wall and dielectric (non-conducting) core are written in vector form as follows: \boldsymbol{E} \times\left.\hat{\mathbf{n}}\right|_S=0 \text { and }\left.\boldsymbol{B} \cdot \hat{\mathbf{n}}\right|_S=0, with S representing the boundary surface between the conductor and dielectric of the waveguide and \hat{\mathbf{n}} the unit vector normal to surface S. Thus, just inside the waveguide core only normal component of {E} and tangential component of \mathcal{B} can exist and moreover, inside the perfect conductor there are no electric and magnetic fields.
In scalar form we express the boundary conditions as: {E}_{\text {tang }} \mid_S=0 \text { and } \mathcal{B}_{\text {norm }} \mid_S= 0, where {E}_{\text {tang }} \mid_S \text { and } \mathcal{B}_{\text {norm }} \mid_S are the tangential component of {E} and normal component of \mathcal{B}, respectively, at the boundary surface. Furthermore, {E}_{\text {tang }} \mid_S , the tangential component of \boldsymbol{E} \text {, is actually given as }{E}_z \mid_S \text { and } \mathcal{B}_{\text {norm }} \mid_S, the normal component of \mathcal{B}, is given as \left.\left(\partial \mathcal{B}_z / \partial n\right)\right|_S . The general boundary conditions on \boldsymbol{E} \text { and } \boldsymbol{B} can thus be summarized as follows

\boldsymbol{E} \times\left.\hat{\mathbf{n}}\right|_S=0 \quad \text { or }\left.\quad {E}_{\text {tang }}\right|_S=0 \quad \text { or }\left.\quad {E}_z\right|_S=0         (13.17)

and

\left.\boldsymbol{B} \cdot \hat{\mathbf{n}}\right|_S=0 \quad \text { or } \quad \mathcal{B}_{\text {norm }} \mid_S=0 \quad \text { or }\left.\quad \frac{\partial \mathcal{B}_z}{\partial n}\right|_S=0 \text {. }         (13.18)

The boundary conditions imposed on {E}_z\ and\ \mathcal{B}_z differ from one another and in general cannot be satisfied simultaneously. Therefore, the transverse fields inside a uniform waveguide are divided into two distinct modes: transverse magnetic (TM) and transverse electric (TE) with the following characteristics:

(1) In the TM mode, the magnetic field \mathcal{B}_z in the direction of propagation is zero everywhere and the boundary condition on {E}_z is given by (13.17).
(2) In the TE mode, the electric field {E}_z in the direction of propagation is zero everywhere and the boundary condition on \mathcal{B}_z is given by (13.18).

(c) The propagation of microwaves through a uniform EM waveguide is governed by four Maxwell equations and appropriate boundary conditions. The four Maxwell equations for electric field {E} and magnetic field \mathcal{B} (in general differential form on the left and in differential form suitable for use with waveguides on the right, accounting for absence of charges and currents resulting in charge density in vacuum ρ = 0 and current density in vacuum j = 0) are given as follows:

(1) Maxwell–Gauss equation (also known as Gauss law of electricity)

\nabla \cdot {E}=\frac{\rho}{\varepsilon_0}           (13.19)

\nabla \cdot {E}=0          (13.20)

(2) Maxwell–Gauss law (also known as Gauss law of magnetism)

\nabla \cdot \boldsymbol{B}=0        (13.21)

\nabla \cdot \mathcal{B}=0           (13.22)

(3) Maxwell–Faraday equation (also known as Faraday law of induction)

\nabla \times \boldsymbol{E}=-\frac{\partial \mathcal{B}}{\partial t}            (13.23)

\nabla \times {E}=-\frac{\partial \mathcal{B}}{\partial t}            (13.24)

(4) Maxwell–Ampère equation (also known as Ampère circuital law)

\nabla \times \boldsymbol{B}=\mu_0 \mathbf{j}+\frac{1}{c^2} \frac{\partial {E}}{\partial t}            (13.25)

\boldsymbol{\nabla} \times \boldsymbol{B}=\frac{1}{c^2} \frac{\partial {E}}{\partial t}            (13.26)

Applying the curl vector operator (∇×) on (13.24) and using the vector identiy

\nabla \times \nabla \times \mathbf{A}=\nabla \nabla \cdot \mathbf{A}-\nabla^2 \mathbf{A}             (13.27)

where

A is an arbitrary vector function,
∇ is the gradient vector operator often labeled as grad,
∇· is the divergence vector operator often labeled as div,
∇² is the vector Laplacian operator where \nabla^2 \equiv \Delta=\nabla \cdot \nabla \equiv \text { divgrad },

results in the following expression linking electric field vector {E} and magnetic field vector \mathcal{B}.

\nabla \times \nabla \times {E}=\nabla \nabla \cdot {E}-\nabla^2 {E}=-\frac{\partial}{\partial t} \nabla \times \mathcal{B}           (13.28)

which, after inserting (13.20) and (13.26), evolves into a 3-dimensional linear partial differential wave equation of the second order in four variables (3 spatial and 1 temporal) for the electric field vector {E}.

\nabla^2 {E}=\frac{1}{c^2} \frac{\partial^2 {E}}{\partial t^2}           (13.29)

(d) Applying the curl vector operator (∇×) on (13.26) and using the vector identity (13.27) results in the following expression linking magnetic field vector \mathcal{B} and electric field vector {E}.

\boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \boldsymbol{B}=\nabla \nabla \cdot \mathcal{B}-\nabla^2 \boldsymbol{B}=-\frac{\partial}{\partial t} \nabla \times {E},           (13.30)

which, after inserting (13.22) and (13.24), evolves into a 3-dimensional partial differential wave equation of the second order in four variables (3 spatial and 1 temporal) for the magnetic field vector \mathcal{B}.

\nabla^2 \boldsymbol{B}=\frac{1}{c^2} \frac{\partial^2 \boldsymbol{B}}{\partial t^2}           (13.31)