Question 13.6.Q1: Wave properties of photons and matter, such as wavelength λ,......

(a) Dispersion relationship for a plane EM wave in vacuum is derived from the basic relationship ν = c/λ for photons in vacuum without any constraints imposed by either boundary conditions or interaction with media. Multiplication of both sides of ν = c/λ with 2π results in the following simple (ω,k) dispersion relationship

2 \pi v=\frac{2 \pi}{\lambda} c \text { or } \omega=c k             (13.142)

since 2πν = ω and 2π/λ = k. The dispersion relationship (13.142) states that angular frequency ω of a photon is proportional to its wave number k with speed of light c in vacuum the proportionality constant.
From (13.142) it follows that the phase velocity υ_{ph}, defined as the ratio ω/k, is equal to speed of light c in vacuum and so is the group velocity υ_{gr}, defined as dω/dk. Thus, for a plane EM wave in vacuum, we have both υ_{ph}\ and\ υ_{gr} equal to a constant, the speed of light c in vacuum.

v_{\mathrm{ph}}=\frac{\omega}{k}=c \quad \text { and } \quad v_{\mathrm{gr}}=\frac{\mathrm{d} \omega}{\mathrm{d} k}=c              (13.143)

(b) Dispersion (ω,k_z) relationship, cutoff frequency ω_c, phase velocity υ_{ph}, and group velocity υ_{gr} for a uniform rectangular EM waveguide are derived from the wave equations for the z components of the electric field E and magnetic field B in conjunction with boundary conditions on the tangential component of E and normal component of B in a rectangular uniform EM waveguide. The wave equations for E and B are derived from Maxwell equations in vacuum [see (T13.13) and (T13.16), respectively and Prob. 274].
The wave equations for {E}_z\ and\ \mathcal{B}_z are given as 3-dimensional linear partial differential equation of the second order in four variables (3 spatial and one temporal) in the following form

\nabla^2 \eta(x, y, z, t)=\frac{1}{c^2} \frac{\partial^2 \eta(x, y, z, t)}{\partial t^2}           (13.144)

where η(x,y,z,t) stands for {E}_z(x,y,z,t) and \mathcal{B}_z(x,y,z,t) components of E and B, respectively
The {E}_z\ and\ \mathcal{B}_z components are the important solutions obtained from wave equations (13.144). The other components \left({E}_x, {E}_y, \mathcal{B}_x, \text { and } \mathcal{B}_y\right) are determined using {E}_z\ and\ \mathcal{B}_z solutions of (13.144) in conjunction with the four Maxwell equations for free space (see Prob. 277).

The most common approach to solving the wave equation (13.144) for η(x,y,z,t) is to apply the method of separation of variables. The method was described in detail in Prob. 276 and here only the important points will be summarized:

(1) First, the time factor is isolated by defining η(x,y,z,t) as a product of two functions: [φ(x,y,z,t) and T (t)] and inserting the product into (13.144) to get

\frac{\nabla^2 \phi}{\phi}=\frac{1}{c^2} \frac{\partial^2 T}{\partial t^2}=-k^2,         (13.145)

where k² is a positive constant and k is called the free space wave number or free space propagation coefficient. The solution for T (t) is T (t) ∝ e^{−iωt} where ω = ck with ω the angular frequency of the RF wave.

(2) Next, function φ(x,y,z) is defined as product of three functions φ(x,y,z) = X(x)Y(y)Z(z) and we now express the Helmholtz equation of (13.145) as

-\frac{1}{X} \frac{\partial^2 X}{\partial x^2}-\frac{1}{Y} \frac{\partial^2 Y}{\partial y^2}=\frac{1}{Z} \frac{\partial^2 Z}{\partial z^2}+k^2=\gamma^2           (13.146)

From the right hand side of (13.146) we get the following Helmholtz equation for Z(z)

\frac{\partial^2 Z}{\partial z^2}+k_z^2 Z=0            (13.147)

with solution for Z(z) given as Z(z) \propto e^{i k_z z}, \text { where } \gamma^2 is a constant and k_z is the waveguide wave number or waveguide propagation coefficient defined as k_z^2=k^2-\gamma^2 for plane wave propagation along the z axis of the Cartesian coordinate system.

(3) The left side of (13.146) now reads

\frac{1}{X} \frac{\partial^2 X}{\partial x^2}+\frac{1}{Y} \frac{\partial^2 Y}{\partial y^2}+\gamma^2=\frac{1}{X} \frac{\partial^2 X}{\partial x^2}+\frac{1}{Y} \frac{\partial^2 Y}{\partial y^2}+\left(k^2-k_z^2\right)=0          (13.148)

and can be separated into two Helmholtz equations given as follows

\frac{\partial^2 X}{\partial x^2}+k_x^2 X=0 \quad \text { and } \quad \frac{\partial^2 Y}{\partial y^2}+k_y^2 Y=0            (13.149)

with the provision that

k_x^2+k_y^2=k^2-k_z^2=\gamma^2 \quad \text { or } \quad k_x^2+k_y^2+k_z^2=k^2=\frac{\omega^2}{c^2}            (13.150)

Equation (13.150) is the so-called dispersion relationship (ω,k_z) linking the RF frequency ω with the propagation coefficient k_z in an evacuated uniform EM waveguide. The dispersion relationship (13.150) is of importance in waveguide theory, since it defines the propagation coefficient k_z for a given frequency ω and a given mode of the RF wave. Moreover, it enables determination of RF frequencies that can propagate through a given waveguide as well as the cutoff frequency ω_c, phase velocity υ_{ph}, and group velocity υ_{gr} of RF waves, based on boundary conditions and waveguide geometry.

(4) Solutions to the two equations in (13.149) as well as coefficients k_x\ and\ k_y are determined from the boundary conditions that are, for {E}\ and\ \mathcal{B} in a rectangular waveguide of sides a and b, respectively, given as follows:

(i) The general Dirichlet boundary condition on the tangential component of electric field {E}_t \mid_S=0 is for a rectangular uniform EM waveguide expressed as

\left.{E}_z\right|_{x=0}=\left.{E}_z\right|_{x=a}=0 \quad \text { and }\left.\quad {E}_z\right|_{y=0}=\left.{E}_z\right|_{y=b}=0 \text {, }           (13.151)

resulting in the following solutions for X(x) and Y(y)

X(x)=\sin k_x x=\sin \frac{m \pi}{a} x \quad \text { and } \quad Y(y)=\sin k_y y=\sin \frac{n \pi}{b} y \text {, }            (13.152)

since from the boundary conditions (13.151) it follows that k_x=\frac{m \pi}{a} and k_y=\frac{n \pi}{b} , where m and n are integers ranging from 1 to ∞ where the lowest value is 1 rather than 0 to avoid nontrivial solutions.

(ii) The general Neumann-type boundary condition on normal component of magnetic field B · \hat n |_S = 0 is for a rectangular uniform EM waveguide expressed as

(ii) The general Neumann-type boundary condition on normal component of magnetic field \left.\boldsymbol{B} \cdot \hat{\mathbf{n}}\right|_S=0 is for a rectangular uniform EM waveguide expressed as

\left.\frac{\partial \mathcal{B}_z}{\partial x}\right|_{x=0}=\left.\frac{\partial \mathcal{B}_z}{\partial x}\right|_{x=a}=0 \quad \text { and }\left.\quad \frac{\partial \mathcal{B}_z}{\partial y}\right|_{y=0}=\left.\frac{\partial \mathcal{B}_z}{\partial y}\right|_{y=b}=0,        (13.153)

resulting in the following solutions for X(x) and Y(y)

X(x)=\cos k_x x=\cos \frac{m \pi}{a} x \quad \text { and } \quad Y(y)=\cos k_y y=\cos \frac{n \pi}{b} y \text {, }          (13.154)

since from the boundary conditions (13.153) it follows that k_x=\frac{m \pi}{a} and k_y=\frac{n \pi}{b} , where m and n are integers ranging from 0 to ∞.

(5) We now take a closer look at the dispersion relationship (13.150), insert the expressions for k_x=m \pi / a \text { and } k_y=n \pi / b from (13.152) and (13.154) into (13.150), and obtain the following dispersion relationship for an evacuated rectangular uniform EM waveguide

k_z=\sqrt{1-\frac{\omega^2}{c^2}-k_x^2-k_y^2}=\frac{1}{c} \sqrt{\omega^2 c^2\left(k_x^2+k_y^2\right)}=\frac{1}{c} \sqrt{\omega^2-\pi^2 c^2\left[\frac{m^2}{a^2}+\frac{n^2}{b^2}\right]},              (13.155)

where the two integers m and n specify the mode of transverse electric and magnetic fields in a waveguide; however, they do not uniquely specify the frequency, since boundaries are in effect only in transverse directions x and y but not in the longitudinal direction of wave propagation along the z axis.

(c) Cutoff frequency ω_c, phase velocity υ_{ph}, and group velocity υ_{gr} for RF waves propagating in a given rectangular uniform EM waveguide are determined as follows:

(1) Cutoff frequency ω_c for a given RF mode. From (13.155) we note that for an RF wave to propagate without attenuation, the propagation constant k_z must be real. Thus, \omega^2 / c^2 \text { must exceed } k_x^2+k_y^2=\pi\left[(m / a)^2+(n / b)^2\right], i.e.,

\frac{\omega}{c}>\pi \sqrt{\frac{m^2}{a^2}+\frac{n^2}{b^2}}           (13.156)

and, as a consequence, each mn mode of a waveguide is associated with a minimum frequency, called cutoff frequency \left(ω_c\right)_{mn} of the mn mode and defined as

\left(\omega_{\mathrm{c}}\right)_{m n}=c \sqrt{\left(\frac{m \pi}{a}\right)^2+\left(\frac{n \pi}{b}\right)^2} .            (13.157)

Inserting (13.157) into (13.155), we now get the following equation for the propagation coefficient k_z.

k_z=\sqrt{\frac{\omega^2}{c^2}-k_x^2-k_y^2}=\frac{1}{c} \sqrt{\omega^2-\pi^2 c^2\left[\frac{m^2}{a^2}+\frac{n^2}{b^2}\right]}=\frac{1}{c} \sqrt{\omega^2-\left(\omega_c\right)_{m n}^2} .            (13.158)

Equation (13.158) can also be expressed in the canonical form of a hyperbola given as

\frac{\omega^2}{\left(\omega_{\mathrm{c}}\right)_{m n}^2}-\frac{c^2}{\left(\omega_{\mathrm{c}}\right)_{m n}^2} k_z^2=1          (13.159)

where the cutoff frequency \left(ω_c\right)^2 _{mn} is the distance between the center C and apex A of the hyperbola. However, in contrast to (13.158) and (13.159), the dispersion relationship for an EM waveguide is most often presented as

\omega^2=\left(\omega_{\mathrm{c}}\right)_{m n}^2+c^2 k_z^2 \quad \text { or } \quad \omega= \pm \sqrt{\left(\omega_{\mathrm{c}}\right)_{m n}^2+c^2 k_z^2} \text {. }         (13.160)

(2) Phase velocity υ_{ph} for a rectangular waveguide is defined as the ratio ω/k_z and represents the speed with which one would need to travel along the waveguide axis in order to stay in phase with the RF wave

From (13.161) we note that υ_{ph} ≥ c and that the dynamic range for the phase velocity v_{\mathrm{ph}} \text { is from } c \text { for } \omega \rightarrow \infty \text { to } \infty \text { for } \omega \rightarrow\left(\omega_{\mathrm{c}}\right)_{m n} \text {, i.e., } c \leq v_{\mathrm{ph}} \leq \infty \text {. }
Thus, phase velocity υ_{ph} of an RF wave in a rectangular uniform waveguide always exceeds the speed of light c in vacuum and approaches c as ω → ∞.
(3) Group velocity υ_{gr} is the speed of propagation of energy and information along the waveguide and is defined as dω/dk_z.

From (13.162) we note that υ_{gr} ≤ c and that the dynamic range of the group velocity is from 0 for ω → \left(ω_c\right)_{mn} to c for ω → ∞, i.e., 0 ≤ υ_{gr} ≤ c. Thus, group velocity υ_{gr}, which represents the velocity of energy and signal propagation in uniform rectangular EM waveguide, is always less than the speed of light c in vacuum and approaches c as ω → ∞.

(d) Figure 13.7 shows a sketch of the hyperbolic dispersion relationship (ω,k_z) with k_z plotted on the abscissa axis and ω on the ordinate axis for a uniform rectangular EM waveguide. The asymptotes to the hyperbola form an angle of arctan c with the k_z axis. For an arbitrary point P on the hyperbola one can determine the phase velocity υ_{ph} and group velocity υ_{gr} as follows:

(1) Phase velocity υ_{ph}: Connect point P with the origin O of the coordinate system (note: point O corresponds to center C of the dispersion hyperbola). The angle between the \overline { PO } line and the +k_z abscissa axis is labeled as \alpha_{\mathrm{ph}} and the tangent of this angle is the phase velocity

v_{\mathrm{ph}}=\tan \alpha_{\mathrm{ph}}=\frac{\omega}{k_z}          (13.163)

(2) Group velocity υ_{gr}: Draw a tangent to dispersion curve at point P. The angle between the tangential line through point P and the k_z axis is labeled as α_{gr} and the trigonometric tangent of this angle is equal to group velocity

v_{\mathrm{gr}}=\tan \alpha_{\mathrm{gr}}=\frac{\mathrm{d} \omega}{\mathrm{d} k_z} .         (13.164)

Figure 13.8 shows a sketch of phase velocity υ_{ph} [see (13.161)] and group velocity υ_{gr} [see (13.162)] both normalized to speed of light c in vacuum and plotted against frequency ω normalized to cutoff frequency ω_c.