Question 13.8.Q1: The theory of waveguides used for charged particle accelerat......
The theory of waveguides used for charged particle acceleration in linear accelerators (linacs) is very complex and relies on many approximations and empirical relationships. However, the acceleration waveguide theory is based solidly on the theory of uniform circular EM waveguides that is well understood, relatively simple, and governs the transmission of microwaves in waveguides. Therefore, each acceleration waveguide has an equivalent circular transmission waveguide that provides a rough guidance for determination of parameters governing electron acceleration in acceleration waveguides.
(a) Microwaves used for acceleration of electrons in standard clinical linacs used for radiotherapy are produced in an RF driver that operates at a frequency ν of 2856 MHz. For microwaves of ν = 2856 MHz and a uniform circular EM waveguide equivalent to an acceleration waveguide of core radius a = 5.25 cm state or calculate:
(1) Wavelength λ and wave number k.
(2) Special RF mode used for electron acceleration in acceleration waveguide.
(3) Cutoff frequency ν_c for the special RF mode used for electron acceleration.
(4) Cutoff frequency ν_c of the acceleration waveguide with core radius a = 5.25 cm.
(5) Parameter γ_n of the equivalent transmission waveguide for the special mode used for electron acceleration of (2).
(6) Parameter γ_n of equivalent transmission waveguide for the dominant special mode.
(7) Wave number (waveguide propagation coefficient) k_g for the special mode used for electron acceleration of (2).
(b) Assume that 2856 MHz microwaves used for electron acceleration in an acceleration waveguide of radius a = 5.25 cm are propagating through an equivalent uniform EM waveguide. For the uniform circular EM waveguide of radius 5.25 cm calculate and plot:
(1) Dispersion relationship (ω vs k_g).
(2) Point P on the ω vs k_g dispersion curve.
(3) Phase velocity υ_{ph} at point P.
(4) Group velocity υ_{gr} at point P.
(c) Show that for particle acceleration with microwaves in a waveguide the particle velocity υ_{part} must be approximately equal to the phase velocity υ_{ph} of the microwaves that are used for particle acceleration in the waveguide.
(d) Explain why uniform EM waveguides are not suitable for charged particle acceleration.
(a) In this problem, the acceleration waveguide has a core radius a of 5.25 cm. Its equivalent uniform waveguide also has a core radius of 5.25 cm and, despite being significantly simpler, provides several useful basic parameters of the acceleration waveguide.
(1) Wavelength λ and wave number k for microwaves with frequency ν = 2856 MHz or angular frequency ω = 2πν = 2π\times(2856 MHz) = 17936 MHz
\lambda = \frac{c}{v} = \frac{3 \times 10^8 m \cdot s^{-1}}{2.856 \times 10^9 s^{-1}} = 0.105 m = 10.5 cm (13.246)
and
k=\frac{2 \pi}{\lambda}=\frac{2 \pi \nu}{c}=\frac{\omega}{c}=\frac{2 \pi}{0.105 \mathrm{~m}} 59.8 \mathrm{~m}^{-1}, (13.247)
where c is the speed of light in vacuum \left(3\times 10^8\ m/s\right).
(2) Electron acceleration is carried out in an acceleration waveguide with microwaves of ν = 2856 MHz (λ = 10.5 cm) propagating in the transverse magnetic (TM_{01}) special mode with m = 0 and n = 1. This mode has several useful features for charged particle acceleration, such as: (i) Z component of magnetic field B_z is zero everywhere in the waveguide core, (ii) E_z \neq 0 on the central axis of the waveguide enabling particle acceleration along waveguide z axis, and (iii) \left. {E}_z\right|_{r=a}=0 at the boundary between the waveguide core and waveguide wall at r = a (Dirichlettype boundary condition).
(3) Cutoff frequency \left(v_{\mathrm{c}}^{\mathrm{TM}}\right) \text { for the } \mathrm{TM}_{01} special mode used for electron acceleration is the same for both the acceleration waveguide and its equivalent uniform waveguide and calculated as
or
\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}=c \frac{x_{01}}{a}=2 \pi\left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{01} 2 \pi \times(2188 \mathrm{MHz})=13741 \mathrm{MHz}, (13.249)
where x_{01} = 2.405 is the first zero (root) of the Bessel function of zero order (see Prob. 281).
(4) The cutoff frequency ν_c of the waveguide is by definition equal to the cutoff frequency of the dominant (lowest) special mode propagating through the waveguide. For a uniform circular EM waveguide the dominant mode is the TE_{11} mode and its cutoff frequency \left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{11} is defined as the waveguide cutoff frequency ν_c and determined from
or
\left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}=2 \pi\left(v_{\mathrm{c}}^{\mathrm{TE}}\right)_{11}=2 \pi \times(1675 \mathrm{MHz})=10519 \mathrm{MHz} (13.251)
where y_{01} = 1.841 is the first zero (root) of the first derivative of the Bessel function of zero order (see Prob. 281). Note that the dominant mode in the circular waveguide (TE_{11}) cannot be used for particle acceleration because in the TE_{mn} special modes E_z = 0 everywhere in the waveguide core and E_z \neq 0 is required for charged particle acceleration.
(5) Parameter γ_n for the TM_{01} mode and microwaves of frequency ν = 2856 MHz
\gamma_n=\frac{x_{01}}{a}=\frac{2.405}{5.25 \times 10^{-2} \mathrm{~m}}=45.8 \mathrm{~m}^{-1} (13.252)
(6) The dominant mode in the waveguide is the TE_{11} mode and the parameter γ_n for this mode is given as follows
\gamma_n=\frac{y_{11}}{a}=\frac{1.841}{5.25 \times 10^{-2} \mathrm{~m}}=35.1 \mathrm{~m}^{-1} (13.253)
(7) Waveguide propagation coefficient k_g for 2856 MHz microwaves and TM_{01} special mode
k_{\mathrm{g}}=\sqrt{k^2-\gamma_n^2}=\sqrt{(59.8 \mathrm{~m})^2-(45.8 \mathrm{~m})^2}=38.5 \mathrm{~m}^{-1} \text {. } (13.254)
(b) The basic dispersion relationship for TM_{mn} modes in uniform circular EM waveguides is
\gamma_n^2=\left(\frac{x_{m n}}{a}\right)^2=\frac{\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}^2}{c^2} k^2-k_{\mathrm{g}}^2=\frac{\omega^2}{c^2}-k_{\mathrm{g}}^2, (13.255)
while for TE_{mn} modes it is given as
\gamma_n^2=\left(\frac{y_{m n}}{a}\right)^2=\frac{\left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n}^2}{c^2} k^2-k_{\mathrm{g}}^2=\frac{\omega^2}{c^2}-k_{\mathrm{g}}^2 (13.256)
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. \left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}^2 and \left(\omega_{\mathrm{c}}^{\mathrm{TE}}\right)_{m n}^2 are cutoff angular frequencies for microwaves propagating through the uniform waveguide in TM_{mn}\ and\ TE_{mn} special modes, respectively.
(1) After rearranging the terms in (13.256) we get the standard form of the general dispersion relationship for TM_{mn} modes in circular EM waveguide of radius a = 5.25 cm
\omega^2=\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}^2+c^2 k_{\mathrm{g}}^2 \quad \text { or } \quad \omega=\sqrt{\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{m n}^2+c^2 k_{\mathrm{g}}^2} (13.257)
that, for the \mathrm{TM}_{01} \text { mode with }\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}=2 \pi\left(v_{\mathrm{c}}^{\mathrm{TM}}\right)_{01} from (13.249), is written as
\omega=\sqrt{\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}^2+c^2 k_{\mathrm{g}}^2}=\sqrt{\left(1.3741 \times 10^{10} \mathrm{~Hz}\right)^2+\left(3 \times 10^8 \mathrm{~m} \cdot \mathrm{s}^{-1}\right)^2 \times k_{\mathrm{g}}^2} . (13.258)
A plot of (13.258) in the k_g range from 0 to ±120 m^{−1} corresponding to ω range from 0 to 45 GHz is shown in Fig. 13.11. The shape of the (13.258) curve shows a hyperbola whose vertex V is the cutoff angular frequency \left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}=1.374 \mathrm{GHz} \text { and whose center } \mathrm{C} is at the origin of the \left(k_{\mathrm{g}}, \omega\right) Cartesian coordinate system.
(2) Also shown in Fig. 13.11 is point P representing the \left(k_g = 38.5\ m^{−1},\ ω = 17.936\ GHz\right) point on the diagram corresponding to propagation of ν = 2856 MHz microwaves through the uniform waveguide in the TM_{01} mode.
(3) Phase velocity υ_{ph} at point P is calculated from the basic definition
v_{\mathrm{ph}}=\frac{\omega}{k_{\mathrm{g}}}=\frac{1.7936 \times 10^{10} \mathrm{~s}^{-1}}{38.5 \mathrm{~m}^{-1}}=4.66 \times 10^8 \mathrm{~m} / \mathrm{s} (13.259)
or from the dispersion relationship (13.257) as
\begin{aligned} v_{\mathrm{ph}} & =\frac{c}{\sqrt{1-\left[\frac{\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}}{\omega}\right]^2}}=\frac{c}{\sqrt{1-\left[\frac{13741}{17936}\right]^2}}=\frac{c}{\sqrt{0.413}} \\ & =1.56 c=4.67 \times 10^8 \mathrm{~m} / \mathrm{s}>c\quad (13.260) \end{aligned}or graphically from the graph of Fig. 13.11 recalling that υ_{ph} = \tan α_{ph}.
(4) Group velocity υ_{ph} at point P is calculated from the dispersion relationship as follows
\begin{aligned} v_{\mathrm{gr}} & =\frac{\mathrm{d} \omega}{\mathrm{d} k_{\mathrm{g}}} c \sqrt{1-\left[\frac{\left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01}}{\omega}\right]^2}=c \sqrt{1-\left[\frac{13741}{17936}\right]^2} \\ & =0.642 c=1.93 \times 10^8 \mathrm{~m} / \mathrm{s}<c\quad (13.261) \end{aligned}or graphically from the graph of Fig. 13.11 recalling that υ_{gr} = \tan α _{gr}.
(c) The phase ϕ of the radiofrequency wave propagating in the +z direction that coincides with the central axis of a uniform circular waveguide is expressed as follows
\varphi=k_{\mathrm{g}} z-\omega t (13.262)
where ω is the angular frequency of the RF wave and k_g is the waveguide wave number or propagation coefficient.
The angular frequency ω of the RF wave as seen by a stationary observer (z = const) is given by
\left|\frac{\mathrm{d} \varphi}{\mathrm{d} t}\right|=\left|\frac{\mathrm{d}}{\mathrm{d} t}\left(k_{\mathrm{g}} z-\omega t\right)\right|=\omega . (13.263)
The angular frequency \omega^{\prime} of the RF wave as seen by an observer (or accelerated charged particle) traveling with the RF wave is calculated as follows
\frac{\mathrm{d} \varphi}{\mathrm{d} t^{\prime}}=\omega^{\prime}=\frac{\mathrm{d}}{\mathrm{d} t^{\prime}}\left(k_{\mathrm{g}} z-\omega t\right)=k_{\mathrm{g}} \frac{\mathrm{dz}}{\mathrm{d} t^{\prime}}-\omega \frac{\mathrm{dt}}{\mathrm{d} t^{\prime}}=k_{\mathrm{g}} \frac{\mathrm{dz}}{\mathrm{d} t} \frac{\mathrm{dt}}{\mathrm{d} t^{\prime}}-\omega \frac{\mathrm{dt}}{\mathrm{d} t^{\prime}}=\left(k_{\mathrm{g}} v_{\text {part }}-\omega\right) \frac{\mathrm{dt}}{\mathrm{d} t^{\prime}}, (13.246)
where {t}^\prime is the time measured in the reference frame of the moving observer and the particle velocity υ_{part} is defined as υ_{part} = dz/dt where t is the time measured by stationary observer.
Since dt/d{t}^\prime is the Lorentz factor γ in relativistic physics and the phase velocity υ_{ph} of the RF wave is defined as υ_{ph} = ω/k_g, we can write (13.264) as
\omega^{\prime}=\frac{\mathrm{d} t}{\mathrm{~d} t^{\prime}}\left(k_{\mathrm{g}} v_{\text {part }}-\omega\right)=\gamma\left(k_{\mathrm{g}} v_{\text {part }}-\omega\right)=\gamma \omega\left(\frac{v_{\text {part }}}{v_{\text {ph }}}-1\right) (13.265)
Equation (13.265) is known as the relativistic Doppler effect. We note from (13.265) that, for the accelerated charged particle to continuously see an accelerating field with a constant phase along the central axis of the waveguide, the angular frequency {ω}^\prime in the reference frame of the charged particle must be zero or at least small. The {ω}^\prime = 0 condition is met when in (13.265) we set υ_{part}/υ_{ph} = 1\ or\ υ_{part} = υ_{ph}. Based on this, one concludes that a necessary condition for particle acceleration with electromagnetic fields in the microwave radiofrequency region is that the particle velocity υ_{part} must be approximately equal to the phase velocity υ_{ph} of microwaves used for particle acceleration.
(d) Several conclusions can be reached based on the discussion in sections (a), (b), and (c):
(1) In (c) we showed that, for charged particle acceleration with microwaves, the accelerated charged particle should see a constant RF phase ϕ during the acceleration process. This 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.
(2) In (b) we showed that in a uniform waveguide the phase velocity υ_{ph} of a typical RF wave used in particle acceleration (ν = 2856 MHz) exceeds the speed of light c in vacuum. Since this is true for all uniform EM waveguides in general and since no particle can travel faster than c in vacuum, it is obvious that uniform waveguides, propagating RF waves with υ_{ph} exceeding c, cannot be used for charged particle acceleration.
(3) The necessary condition for charged particle acceleration with microwaves is that particle velocity υ_{part} is equal to the RF phase velocity υ_{ph}. Since υ_{ph} in uniform EM waveguides exceeds c and particle velocity cannot exceed c, it is obvious that means must be used to decrease υ_{ph} of uniform waveguides below c, if RF waves are to be used for particle acceleration. This is what was done when acceleration EM waveguide was developed from uniform circular EM waveguide by loading the latter with disks or irises that slow down the phase velocity of the RF wave to slightly below c enabling the accelerated charged particle to follow the accelerating RF field.
(4) We also note from (a) that the special waveguide mode that fulfills the condition of having electric field in the direction of circular waveguide central axis is the TM_{01} transverse magnetic mode in contrast with the dominant mode of the circular waveguide that is the TE_{11} transverse electric mode that cannot be used for particle acceleration but is very suitable for transmission of microwave power and signals.
