Question 13.9.Q1: During the past three decades linear accelerator (linac) gre......

(a) In a uniform circular EM waveguide of core radius a, condition (i) is satisfied in the simplest manner by the transverse magnetic TM_{01} special mode propagating through a uniform EM waveguide. This mode is the dominant TM_{mn} mode; however, it is not the dominant mode of the waveguide. This distinction belongs to the transverse electric TE_{11} mode that has the lowest cutoff frequency of all modes propagating through the waveguide, but does not have a non-zero electric field component {E}_z in the direction of the waveguide axis. Therefore, the TE_{11} mode cannot be considered for electron acceleration; however, it is used for efficient transmission of microwave power and signals, while the TM_{01} mode is a candidate for use in electron acceleration with radiofrequency (RF) waves, because it fulfills condition (i) above.
The transverse magnetic TM_{01} mode fulfills condition (i) and is characterized by the following attributes:

(1) Magnetic field component \mathcal{B}_z is zero everywhere in the waveguide core (only the transverse component \mathcal{B}_θ is present in the core; \mathcal{B}_r is also zero).
(2) In contrast to the TE_{11} mode where {E}_z = 0 everywhere in the core of the waveguide, in the TM_{01} mode a non-zero electric field component {E}_z is present on the central axis of the waveguide, thereby satisfying condition (i) for particle acceleration.
(3) The {E}_z component of the electric field {E} is governed by the Dirichlet-type boundary condition \left.{E}_z\right|_{r=a}=0 at the r = a boundary between the evacuated waveguide core and the conductive waveguide wall.

(b) Condition (ii) results from the requirement that the accelerated charged particle must continuously see an accelerating electric field in the direction of propagation along the central axis of the acceleration waveguide. This means that the phase ϕ of the {E}_z component of the electric field {E} must remain constant; that is, the angular frequency {ω}^\prime of the RF wave as measured by an observer traveling with the accelerated particle should be zero or at least very small. The {ω}^\prime = 0 condition is fulfilled when the velocity υ_{part} of the accelerated particle is equal to the phase velocity υ_{ph} of the RF wave used in particle acceleration (see Prob. 283).

(c) Condition (i) is fulfilled by the TM_{01} special mode propagating through a uniform circular EM waveguide and condition (ii) simply states that υ_{part} = υ_{ph}. However, condition (iii) requires that v_{\mathrm{ph}} \lesssim c and this immediately excludes the potential use of a uniform waveguide for charged particle acceleration, because the phase velocity υ_{ph} of microwaves propagating in a uniform EM waveguide always exceeds c and approaches c only for ω → ∞.
Since particle velocity cannot be increased above c, the condition υ_{part} = υ_{ph} can be met only by decreasing the phase velocity of the RF wave in the waveguide below c. This entails adding a periodic perturbation into a uniform waveguide of core radius a in the form of disks or irises with circular holes of radius b in the center. At these obstacles RF waves suffer partial reflection and this effectively reduces υ_{ph} of the RF wave propagating through the waveguide. With an appropriate choice of disk separation and radii a and b where b \ll a, wave reflections at the disks can push υ_{ph} down to a level where v_{\mathrm{ph}} \lesssim c, making particle acceleration physically possible.
In contrast to uniform circular EM waveguides (also called transmission EM waveguides) circular EM waveguides suitable for particle acceleration are called acceleration waveguides or disk-loaded waveguides. They are more complicated than uniform waveguides; however, many parameters of acceleration waveguides are similar to those of uniform waveguides of the same radius a, so that data measured for transmission waveguides are routinely used to approximate parameters of acceleration waveguides.

(d) The velocity υ_{en} of energy flow in a waveguide will be determined from the general relationship

\bar{P}=W_{\mathrm{tot}} v_{\mathrm{en}} \quad \text { or } \quad v_{\mathrm{en}}=\frac{\bar{P}}{W_{\mathrm{tot}}} \text {, }             (13.266)

where

\bar{P} is the mean power flowing through a transverse cross section A of the waveguide.
W_{tot} is the total EM energy stored per unit length in the waveguide.

Mean power \bar{P} is related to the Poynting vector S that represents energy flow with dimensions of energy/(area×time) or power/area. Mean power \bar{P} is thus determined by integrating the Poynting vector S over the transverse cross section A of the waveguide

\bar{P}=\iint_A \overline{\mathbf{S}} \mathrm{d} \mathbf{A}=\frac{1}{2 \mu_0} \operatorname{Re} \iint_A \boldsymbol{E} \times \boldsymbol{B}^* \mathrm{~d} \mathbf{A}=\frac{1}{2 \mu_0} \operatorname{Re} \iint_A {E}_{\mathrm{T}} \mathcal{B}_{\mathrm{T}}^* \mathrm{~d} A            (13.267)

where E and B are the electric and magnetic field, respectively, and E_T and B_T are the transverse components (perpendicular to direction of propagation as well as to one another) of E and B, respectively. The factor of \frac{1}{2} in (13.267) arises from the time average over one complete period, Re stands for “real part”, and the asterisk (*) indicates complex conjugate.

The time average of stored energy W_{tot} per unit length of the waveguide has two components: electric W_{el} and magnetic W_{mag} given as follows

W_{\mathrm{el}}=\frac{1}{4} \varepsilon_0 \iint_A {E}_{\mathrm{T}} {E}_{\mathrm{T}}^* \mathrm{~d} A         (13.268)

and

W_{\text {mag }}=\frac{1}{4 \mu_0} \iint_A \mathcal{B}_{\mathrm{T}} \mathcal{B}_{\mathrm{T}}^* \mathrm{~d} A,          (13.269)

the extra factor of \frac{1}{2} in (13.268) and (13.269) arises from the time average over one complete period, and ε_0 and μ_0 are the electric and magnetic constant, respectively.
The stored energy per unit length of the waveguide W_{tot} is given as the sum of the two components: W_{el}\ and\ W_{mag}.

W_{\text {tot }}=W_{\mathrm{el}}+W_{\mathrm{mag}}=\frac{1}{4} \varepsilon_0 \iint_A {E}_{\mathrm{T}} {E}_{\mathrm{T}}^* \mathrm{~d} A+\frac{1}{4 \mu_0} \iint_A \mathcal{B}_{\mathrm{T}} \mathcal{B}_{\mathrm{T}}^* \mathrm{~d} A=\frac{1}{2 \mu_0} \iint_A \mathcal{B}_{\mathrm{T}} \mathcal{B}_{\mathrm{T}}^* \mathrm{~d} A,           (13.270)

where in the last term of (13.270) we simplified the expression for W_{tot} recognizing that the two components W_{el}\ and\ W_{mag} are equal giving W_{tot} = 2W_{mag}.
Inserting (12.267) and (13.270) into (13.265) results in the following general expression for the velocity of energy flow in a uniform EM waveguide

v_{\mathrm{en}}=\frac{\bar{P}}{W_{\text {tot }}}=\frac{\frac{1}{2 \mu_0} \operatorname{Re} \iint_A {E}_{\mathrm{T}} \mathcal{B}_{\mathrm{T}}^* \mathrm{~d} A}{\frac{1}{2 \mu_0} \iint_A \mathcal{B}_{\mathrm{T}} \mathcal{B}_{\mathrm{T}}^* \mathrm{~d} A} .            (13.271)

We now use the general expression (13.271) to calculate the velocity of energy flow for the transverse magnetic TM_{01} mode of microwaves propagating in a uniform circular EM waveguide of radius a. The TM_{01} mode is the dominant TM_{mn} mode in a circular EM waveguide and is characterized by:

(i) \mathcal{B}_z = 0 everywhere in the waveguide core.
(ii) Dirichlet-type boundary condition \left.{E}_z\right|_{r=a}=0 .
(iii) Non-zero z component of electric field \left({E}_z \neq 0\right) on the axis of the waveguide and as such under appropriate circumstances can be used for charged particle acceleration.

In general, the {E}_z\ and\ \mathcal{B}_z components of {E} and\ \mathcal{B} are determined from wave equations and the remaining components \left({E}_r, {E}_θ ,\ \mathcal{B}_r,\ and\ \mathcal{B}_θ \right)\ of  {E}\ and\ \mathcal{B} are determined from Maxwell equations. In Prob. 282(e) we derived the following expressions for the z components as well as the transverse components of {E}\ and\ \mathcal{B} of the TM_{01} mode in a uniform circular EM waveguide of radius a

{E}_r=-i k_{\mathrm{g}}\left(\frac{a}{x_{01}}\right) {E}_{01} J_1\left(\frac{x_{01}}{a} r\right) e^{i \varphi},            (13.272)

{E}_\theta=0,           (13.273)

{E}_z={E}_{01} J_0\left(\frac{x_{01}}{a} r\right) e^{i \varphi},          (13.274)

\mathcal{B}_r=0           (13.275)

\mathcal{B}_\theta=-i \frac{\omega}{c^2}\left(\frac{a}{x_{01}}\right) {E}_{01} J_1\left(\frac{x_{01}}{a} r\right) e^{i \varphi},           (13.276)

\mathcal{B}_z=0 .          (13.277)

For the TM_{01} mode the mean power \bar{P} of (13.267) and energy stored per unit distance W_{tot} of (13.271) are calculated as follows

and

After inserting (13.278) and (13.279) into (13.266) we get, for the TM_{01} mode, the following expression for the velocity υ_{en} of energy flow in a uniform circular EM waveguide of radius a

v_{\mathrm{en}}=\frac{\bar{P}}{W_{\text {tot }}}=\frac{\frac{\pi k_{\mathrm{g}} \omega}{\mu_0 c^2}\left(\frac{a}{x_{01}}\right)^2 {E}_{01}^2 \int_0^a J_1^2\left(\frac{x_{01}}{a} r\right) r \mathrm{~d} r}{\frac{\pi \omega^2}{\mu_0 c^4}\left(\frac{a}{x_{01}}\right)^2 {E}_{01}^2 \int_0^a J_1^2\left(\frac{x_{01}}{a} r\right) r \mathrm{~d} r}=\frac{k_{\mathrm{g}} c^2}{\omega}           (13.280)

Several observations are now possible after a closer look at the result v_{\mathrm{en}}=\frac{k_{\mathrm{g}} c^2}{\omega}:

(1) Since by definition v_{\mathrm{ph}}=\frac{\omega}{k_{\mathrm{g}}} \text { and } v_{\mathrm{ph}} \geq c \text {, we note that } v_{\mathrm{en}}=\frac{c^2}{v_{\mathrm{ph}}} \leq c \text {. }

(2) Since v_{\mathrm{ph}} v_{\mathrm{gr}}=c^2 \text {, we note from (13.280) that } v_{\mathrm{en}}=v_{\mathrm{gr}} \text {. }

(3) Since k_{\mathrm{g}}=\frac{\sqrt{\omega^2-\omega_{\mathrm{c}}^2}}{c} \text {, we note that } v_{\mathrm{en}}=c \sqrt{1-\frac{\omega_{\mathrm{c}}^2}{\omega^2}}=v_{\mathrm{gr}}

Thus, the velocity υ_{en} of energy flow through the waveguide is equal to the group velocity υ_{gr} of microwave propagation in the waveguide.