Question 13.6.Q3: The theory of uniform circular electromagnetic (EM) waveguid......

(a) Parameters of uniform circular evacuated EM waveguide (see Prob. 276):

(1) Dispersion relationship for a uniform circular EM waveguide is in general form written as

\gamma_n^2=k^2-k_{\mathrm{g}}^2          (13.179)

where k is the free space wave number (k = ω/c), k_g is the wave number or propagation coefficient of the circular waveguide, and γ_n is a constant determined from E_z\ and\ B_z solutions to wave equations in conjunction with the boundary conditions on E_z\ and\ B_z for a circular EM waveguide.
The z components E_z\ and\ B_z of the electric field E and magnetic field B are written, respectively, in general form as a double series

and

where A_{mn},\ B_{mn},\ C_{mn},\ and\ D_{mn} are constants that can be determined from initial conditions. Parameter γ_n in the argument of the Bessel function J_m\left(γ_nr\right) in (13.180) and (13.181) is determined from the boundary conditions on E_z\ and\ B_z. Since these are generally different, they cannot be applied simultaneously and the two fields are split into two special categories or modes [transverse magnetic (TM) and transverse electric (TE)], characterized as follows:

(i) For the TM_{mn}\ modes,\ B_z = 0 everywhere in the waveguide core and E_z is governed by the Dirichlet-type boundary condition \left. {E}_z\right|_{r=a}=0 which specifies that E_z = 0 at the boundary between the waveguide core and the conducting wall of the waveguide. The \left. {E}_z\right|_{r=a}=0 boundary condition results in the following solution for γ_n of (13.180)

\left. {E}_z\right|_{r=a}=\left.J_m\left(\gamma_n r\right)\right|_{r=a}=J_m\left(\gamma_n a\right)=0 \quad \text { resulting in } \quad \gamma_n=\frac{x_{m n}}{a} \text {, }                (13.182)

where x_{mn} is the n-th zero (root) of the m-th order Bessel function of the first kind \left(J_m\right). Insertion of (13.182) into (13.179) results in the following expression for the TM_{mn} dispersion relationship in the form of ω = f \left(k_g\right).

\omega^2=c^2\left(\frac{x_{m n}}{a}\right)^2+c^2 k_{\mathrm{g}}^2=\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}+c^2 k_{\mathrm{g}}^2 \quad \text { or } \quad \omega=\sqrt{\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}+c^2 k_{\mathrm{g}}^2},              (13.183)

where \left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}=c x_{m n} / a is the cutoff frequency for \mathrm{TM}_{m n} \text { modes at } k_{\mathrm{g}}=0

(ii) For the TE_{mn}\ modes,\ E_z = 0 everywhere in the waveguide core and B_z is governed by the Neumann-type boundary condition \mathrm{d} \mathcal{B}_z /\left.\mathrm{d} r\right|_{r=a}=0 that results in the following solution for γ_n of (13.181)

\left.\frac{\mathrm{d} \mathcal{B}_z}{\mathrm{~d} r}\right|_{r=a}=\left.\frac{\mathrm{d} J_m\left(\gamma_n r\right)}{\mathrm{d} r}\right|_{r=a}=\frac{\mathrm{d} J_m\left(\gamma_n a\right)}{\mathrm{d} r}=0 \text { resulting in } \quad \gamma_n=\frac{y_{m n}}{a},               (13.184)

where y_{mn} is the n-th zero (root) of the first derivative of the m-th order Bessel function of the first kind. Insertion of (13.184) into (13.179) results in the following expression for the TE_{mn} dispersion relationship in the form of ω = f \left(k_g\right).

\omega^2=c^2\left(\frac{y_{m n}}{a}\right)^2+c^2 k_{\mathrm{g}}^2=\left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n}+c^2 k_{\mathrm{g}}^2 \quad \text { or } \quad \omega=\sqrt{\left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n}+c^2 k_{\mathrm{g}}^2},             (13.185)

where \left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n}=c y_{m n} / a is the cutoff frequency for \mathrm{TE}_{m n} \text { modes at } k_{\mathrm{g}}=0 \text {. }

(2) Cutoff frequency \left(ω_c\right)_{mn} is the lowest frequency with which a mode mn can propagate through an EM waveguide. All frequencies exceeding \left(ω_c\right)_{mn} can propagate through the waveguide without attenuation; frequencies below \left(ω_c\right)_{mn} are attenuated and cannot propagate through the waveguide.

(i) Cutoff frequency \left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n} for given transverse magnetic (TM) mode mn is from (13.183) for k_g = 0 expressed as

\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}=c \frac{x_{m n}}{a},           (13.186)

where x_{mn} is the n-th zero (root) of the m-th order Bessel function \left(J_m\right).

(ii) Cutoff frequency \left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n} for given transverse electric (TE) mode mn is from (13.185) for k_g = 0 expressed as

\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}=c \frac{y_{m n}}{a},           (13.187)

where y_{mn} is the n-th zero (root) of the first derivative of the m-th order Bessel function.

(3) Lowest cutoff frequency \left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n} \text { for } \mathrm{TM}_{m n} \text { modes. The lowest } \mathrm{TM}_{m n} mode will occur for m = 0 and n = 1, giving the following expressions for E_z of (13.180)

{E}_z(r, \theta, z, t)= {E}_{z 0} J_0\left(\frac{x_{01}}{a} r\right) e^{i \varphi}= {E}_{z 0} J_0\left(\frac{2.405}{a} r\right) e^{i \varphi}             (13.188)

and for \left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n} of (13.186)

\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}=c \frac{x_{01}}{a}=c \frac{2.405}{a},             (13.189)

where E_{z0} is the electric field amplitude, ϕ is the phase of the wave, x_{01} = 2.405 is the first zero (root) of the zeroth order Bessel function of the first kind \left[J_0(z)\right], c is the speed of light in vacuum, and a is the radius of the uniform circular evacuated EM waveguide.

(4) Lowest cutoff frequency \left(\omega_c^{\mathrm{TE}}\right)_{m n} \text { for } \mathrm{TE}_{m n} \text { modes. The lowest } \mathrm{TE}_{m n} mode will be for m = 1 and n = 1, giving the following expression for B_z of (13.181)

\mathcal{B}_z(r, \theta, z, t)=\mathcal{B}_{z 0} J_1\left(\frac{y_{11}}{a} r\right) e^{i \varphi}=\mathcal{B}_{z 0} J_1\left(\frac{1.841}{a} r\right) e^{i \varphi}             (13.190)

and for \left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n} of (13.187)

\left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}=c \frac{y_{11}}{a}=c \frac{1.841}{a},             (13.191)

where B_{z0} is the magnetic field amplitude, ϕ is the phase of the wave, and y_{11} = 1.841 is the first zero (root) of the derivative of the J_1(z) Bessel function of the first kind.

(5) Using (13.189) and (13.191), the ratio between the lowest cutoff frequency \left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01} for TM modes and the lowest cutoff frequency \left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{11} for TE modes is given as

\frac{\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}}{\left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}}=\frac{2.405}{1.841}=1.306 .           (13.192)

(6) Cutoff frequency ω_c for a given waveguide is defined as the cutoff frequency of the lowest mode mn that can propagate through a waveguide. Thus, the TE_{11} is the mode with the lowest cutoff frequency of all modes in a uniform circular EM waveguide and therefore the cutoff frequency for circular waveguides is from (13.191) given as ω_c = 1.841c/a. Note that the cutoff frequency of a circular EM waveguide is inversely proportional to radius a of the waveguide core.
Circular waveguides for transmission of a given radiofrequency (RF) are usually designed such that, at the given RF, the only mode they transmit is the TE_{11} mode. This means that the cutoff frequencies of all TM modes as well as the cutoff frequencies of all TE modes with the exception of the TE_{11} mode exceed the frequency of the given RF.
Circular waveguides used for particle acceleration are loaded with disks (irises) and designed such that, in addition to the TE_{11} mode, they also transmit the TM_{01} mode to enable particle acceleration.

(b) Figure 13.10 displays five Bessel functions J_m(z) for 0 ≤ m ≤ 4 and z in the range from 0 to 10. Superimposed onto the diagram are zeros (roots) x_{mn} of the five Bessel functions as well as zeros (roots) y_{mn} of the first derivatives of the five Bessel functions.
Roots x_{mn} of a Bessel function J_m(z) occur at points where the Bessel curve crosses the abscissa (z) axis with m designating the order of the Bessel function and n designating the rank of a given root starting with n = 1 for the first non-trivial root as z increases from 0 to ∞.

Roots y_{mn} of the first derivative dJ_m(z)/dz of Bessel function J_m(z) occur at points where J_m(z) exhibits a local maximum or minimum, i.e., where the tangent on the Bessel curve becomes horizontal. In y_{mn} parameter m again is the order of the Bessel function and n is the rank of the root starting with n = 1 for the root with the lowest z value.
Roots x_{mn}\ of\ y_{mn}\ and\ J_m(z)\ and\ dJ_m(z)/dz, respectively, for 0 ≤ m ≤ 4 and 0 ≤ z ≤ 10 estimated to two significant figures from the five curves of Fig. 13.10 are listed in Table 13.4 for use in (c) in calculation of cutoff frequencies \left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n} and \left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n} of uniform circular EM waveguides.

(c) Cutoff frequencies \left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n} \text { and }\left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n} \text { for the } \mathrm{TM}_{m n} \text { and } \mathrm{TE}_{m n} modes, respectively, in a uniform circular EM waveguide of core radius a were calculated from the following expressions derived from (13.186) and (13.187), respectively

\left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}=\frac{1}{2 \pi}\left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}=\frac{c}{2 \pi a} x_{m n}=\xi_a x_{m n}           (13.193)

and

\left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n}=\frac{1}{2 \pi}\left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n}=\frac{c}{2 \pi a} y_{m n}=\xi_a y_{m n},            (13.194)

where x_{mn} is the n-th zero (root) of the m-th order Bessel function and y_{mn} is the n-th zero of the first derivative of the m-th order Bessel function, and ξ_a = c/(2πa) is a constant used in calculation of TM and TE cutoff frequencies in an evacuated waveguide of radius a.
Before embarking on calculation of cutoff frequencies using (13.193) and (13.194) we determine the waveguide constant ξ_a for use in (13.193) to determine TM cutoff frequencies \left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n} and in (13.194) to determine TE cutoff frequencies \left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n}.

\xi_a=\frac{c}{2 \pi a}=\frac{3 \times 10^8 \mathrm{~m} / \mathrm{s}}{2 \pi \times\left(1.05 \times 10^{-2} \mathrm{~m}\right)}=4.55 \times 10^9 \mathrm{~Hz}=4550 \mathrm{MHz}             (13.195)

Next we calculate a set of low-level TM cutoff frequencies using (13.193) in conjunction with (13.195) and x_{mn} data of Table 13.4 and a set of TE cutoff frequencies using (13.194) in conjunction with (13.195) and y_{mn} data of Table 13.4. Results of TM and TE cutoff frequencies calculated for a uniform circular EM waveguide with core radius of 1.05 cm are presented in Table 13.5.
According to Table 13.5:

(1) Five lowest cutoff frequencies \left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n} \text { for } \mathrm{TM}_{m n} modes are: TM_{01} (10920 MHz), TM_{11} (17290 MHz), TM_{21} (23205 MHz), TM_{02} (25025 MHz), and TM_{31} (29120 MHz).

(2) Five lowest cutoff frequencies \left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n} \text { for } \mathrm{TE}_{m n} \text { modes are: } \mathrm{TE}_{11} (8190 MHz), TE_{21} (14105 MHz), TE_{02} (17290 MHz), TE_{31} (19110 MHz), and TE_{41} (24115 MHz).

(3) Combined order of 5 lowest special modes in the waveguide is as follows: TE_{11} (8190 MHz), TM_{01} (10920 MHz), TE_{21} (14105 MHz), TM_{11} (17290 MHz), and TE_{01} (17290 MHz).

(4) The cutoff frequency ω_c = 2πν_c of the waveguide is given by the lowest cutoff frequency of the two special modes (TM and TE) propagating through the waveguide. For the circular EM waveguide in this problem, the lowest cutoff frequency occurs for the \mathrm{TE}_{11} \text { mode with }\left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}=8190 \mathrm{MHz} and we conclude that the cutoff frequency ν_c of the waveguide is 8190 MHz.

(d) The input microwaves to be transmitted through the circular waveguide have a frequency ν = 10^4 MHz in the X band microwave frequency range. This frequency is below the lowest TM cutoff frequency of \left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}=10920 \mathrm{MHz} which means that no TM modes can propagate in this EM waveguide. Of the TE modes only the TE_{11} can propagate, since its cutoff frequency of 8190 MHz is below the input microwave frequency of 10^4 MHz. Thus, the circular EM waveguide with a radius of 1.05 cm allows propagation of 10^4 MHz microwaves only in one mode, the TE_{11} mode with a cutoff frequency \left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}=8190 \mathrm{MHz}. This is another example of waveguide design that allows only a single mode operation.

(e) The phase velocity υ_{ph} and group velocity υ_{gr} of microwaves propagating in a uniform circular EM waveguide with angular frequency ω or frequency ν are calculated from the dispersion relationship (ω,k_g) given in (13.184) as follows

\omega=\sqrt{\left(\omega_{\mathrm{c}}\right)_{m n}^2+c^2 k_{\mathrm{g}}^2} \quad \text { or } \quad k_{\mathrm{g}}=\frac{1}{c} \sqrt{\omega^2-\left(\omega_{\mathrm{c}}\right)_{m n}^2} \text {, }             (13.196)

where \left(\omega_{\mathrm{c}}\right)_{m n}=2 \pi\left(v_{\mathrm{c}}\right)_{m n} is the cutoff frequency of mode mn propagating through the circular EM waveguide and k_g is the propagation coefficient of the waveguide.

(1) In general, the phase velocity υ_{ph} is defined as the ratio between ω of the propagating wave and the associated k_g of the waveguide

v_{\mathrm{ph}}=\frac{\omega}{k_{\mathrm{g}}}=\frac{c \omega}{\sqrt{\omega^2-\left(\omega_{\mathrm{c}}\right)_{m n}^2}}=\frac{c}{\sqrt{1-\frac{\left(\omega_{\mathrm{c}}\right)_{m n}^2}{\omega^2}}}=\frac{c}{\sqrt{1-\frac{\left(v_{\mathrm{c}}\right)_{m n}^2}{v^2}}} .                 (13.197)

In (d) we established that in a circular EM waveguide with core radius a = 1.05 cm microwaves with frequency ν = 10^4 MHz in the X microwave band can only propagate in a \mathrm{TE}_{11} \text { mode that has a cutoff frequency }\left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}=8190 \mathrm{MHz} \text {. } Therefore, the phase velocity υ_{ph} of 10000 MHz microwaves propagating in this waveguide is from (13.197) calculated as follows

v_{\mathrm{ph}}=\frac{c}{\sqrt{1-\frac{\left(\nu_c^{\mathrm{TM}}\right)_{11}^2}{v^2}}}=\frac{c}{\sqrt{1-\left(\frac{8190}{10000}\right)^2}}=1.74 c=5.5 \times 10^8 \mathrm{~m} / \mathrm{s}>c .           (13.198)

(2) Group velocity υ_{gr} is in general defined as the derivative dω/dk_g.

For the cutoff frequency v_{\mathrm{c}}=\left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}=8190 \mathrm{MHz} and microwave frequency of 10^4 Hz propagating through a circular EM waveguide of radius a = 1.05 cm the group velocity υ_{gr} is from (13.199) calculated as

v_{\mathrm{gr}}=c \sqrt{1-\frac{\left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}^2}{v^2}}=c \sqrt{1-\left(\frac{8190}{10000}\right)^2}=0.57 c=1.72 \times 10^8 \mathrm{~m} / \mathrm{s}<c .           (13.200)