Question 13.10.Q1: Uniform electromagnetic (EM) waveguides (also called transmi......

(a) Both types of waveguide: (i) uniform (transmission) waveguide and (ii) accelerating (disk-loaded) waveguide are used for transmission of microwave power; however, in design and purpose there are significant differences between the two waveguide types, such as:

(1) Transmission waveguide is much simpler in design than acceleration waveguide.
(2) Transverse cross section of transmission waveguide is either rectangular or
circular, while cross section of acceleration waveguide is always circular.
(3) Core of transmission waveguide is either evacuated or filled with pressurized
dielectric gas, that of acceleration waveguide is always evacuated.
(4) Transmission waveguide is essentially a pipe of uniform cross section (rectangular or circular); acceleration waveguide is a cylinder loaded with periodic perturbations in the form of disks (irises) that create partitions in the waveguide tube and define sections called cavity in the waveguide.
(5) The phase velocity υ_{ph} of microwaves propagating in a transmission waveguide exceeds the speed of light c in vacuum making charged particle acceleration impossible; the phase velocity of microwaves propagating in acceleration waveguide is slightly less than c to allow charged particle acceleration. The role of disks in acceleration waveguide is to decrease the phase velocity υ_{ph} of a uniform waveguide, where for practical microwave frequencies it always exceeds the speed of light c in vacuum, to a level below c to allow charged particle acceleration.
(6) It is clear that the transmission (uniform) waveguide and the acceleration (disk-loaded) waveguide are related, since the latter slowly evolved from the former. One may state that in view of this relationship each disk-loaded waveguide can be approximated with a uniform waveguide and we can refer to it as an equivalent, yet much simpler, waveguide. This equivalent waveguide has the same core radius as the accelerator waveguide but cannot be used for charged particle acceleration (recall that υ_{ph} > c in uniform waveguide and for particle acceleration we must have υ_{part} ≈ υ_{ph}); however, the equivalent uniform waveguide can serve as a simple pathway toward determination of the basic parameters of the acceleration waveguide.

(b) Figure 13.12 shows schematic diagrams of a uniform (transmission) circular EM waveguide of core radius a and of a disk-loaded (acceleration) EM waveguide of core radius a, radius b of circular hole at the center of the disk, disk-separation (cavity height) d, and disk thickness s.

(c) Figure 13.13 shows the dispersion (ω,k_g) diagrams for an acceleration waveguide (solid curves) and for its equivalent uniform waveguide (dotted curve). The uniform waveguide curve is a hyperbola with only one cutoff frequency ω_c and its pass band ranges in frequency from ω = ω_c to ω = ∞, while its stop band has frequencies from ω = 0 to ω = ω_{c}. The acceleration waveguide features an infinite number of pass bands and stop bands, but only the lowest pass band for ω_{c1} <ω<ω_{c2} in the transverse magnetic TM mode can be used for charged particle acceleration. The lowest cutoff frequency ω_{c1} of the acceleration waveguide in the TM mode is equal to the cutoff frequency ω_c of the equivalent uniform waveguide.

(d) Figure 13.13 depicts dispersion (ω,k_g) diagrams for an acceleration (diskloaded) waveguide (solid curves) and its equivalent (same core radius a) transmission (uniform) waveguide (dotted curve). The dispersion diagram for the equivalent uniform waveguide is a simple hyperbola with its vertex defining the cutoff frequency ω_c and the center of the hyperbola coinciding with the origin of the Cartesian (k_g,ω) coordinate system.

The dispersion diagram for the acceleration waveguide is considerably more complicated than that of its equivalent uniform waveguide. It features an infinite number of pass bands and stop bands that are limited by cutoff frequencies ω_{cN} and Brillouin zones limited by distinct wavenumbers k_g in increments of π/d with d the separation between two successive disks in the waveguide, as shown in Fig. 13.12.
Brillouin zones play an important role in many types of periodic structures in physics and engineering, such as, for example, models governing x-ray, neutron, and electron wave propagation in crystals where the crystal lattice represents a typical periodic structure. The dispersion diagram (E,k) for a crystal can be plotted in the form of energy E against wave number k for: (i) a free electron resulting in a dispersion diagram (E,k) similar to the dispersion diagram (ω,k_g) for a uniform waveguide and (ii) for an electron in a monoatomic linear lattice of lattice constant d, resulting in a dispersion diagram (E,k) similar to the dispersion diagram (ω,k_g) for a disk-loaded EM waveguide.
Charged particle acceleration is carried out only in the first Brillouin zone in the k_g range extending from −π/d to π/d and in the TM_{01} mode with the cutoff angular frequency \left(\omega_{\mathrm{c} 1}^{\mathrm{TM}}\right)_{01} that is identical to the cutoff frequency \left(\omega_{\mathrm{c}}^{\mathrm{TM}}\right)_{01} of the equivalent uniform (transmission) waveguide.
As an RF wave propagates through a disk-loaded waveguide, it is partially reflected at each disk, the reflected fraction depending on the relative magnitudes of the wavelength λ_g and the perturbation parameter (a − b) with a the radius of the uniform waveguide and b the radius of the disk opening, as shown in Fig. 13.12. When radius b is comparable to a, i.e, a-b \ll a, the perturbation caused by the disks is small, the reflection of the radiofrequency wave at the disk is negligible, and the disk-loaded waveguide behaves much like uniform waveguide with radius a.
In general, when \lambda_{\mathrm{g}} \gg(a-b) \text {, corresponding to } k_{\mathrm{g}} \ll(a-b), the fraction of wave reflection at disks is small. The dispersion relationship of the disk-loaded waveguide then tends to that of a uniform waveguide and the cut-off frequency ω_c\ at\ k_g = 0 of the disk-loaded waveguide is identical to that of a uniform waveguide. However, with increasing k_g, corresponding to a decreasing λ_g since k_g = 2π/λ_g, the fraction of the reflected wave at each disk steadily increases, and so does the interference between the incident and reflected wave, until at λ_g = 2d\ or\ k_g = π/d purely stationary waves are setup in each cavity defined by the disk separation d. In this case, the cavities are in resonance, only stationary waves are present in the cavities, and there is no energy propagation possible from one cavity to another. This implies that the group velocity υ_{gr} at point F of Fig. 13.13 is zero \left(υ_{gr} = 0\right) and the tangent to the \left(ω − k_g\right) dispersion relationship at k_g = π/d must be horizontal, in contrast to the uniform waveguide where the tangent to the dispersion relationship is horizontal only at k_g = 0, and then with an increasing k_g its slope steadily rises to its limit of c as k_g → ∞.
For both the uniform as well as the disk-loaded waveguide the group velocity υ_{gr} is zero at the cutoff frequency corresponding to the propagation coefficient k_g = 0 (point A in Fig. 13.13). As k_g increases from 0, the group velocity for uniform waveguide steadily increases until at k_g = ∞ it reaches a value of c. For a diskloaded waveguide, on the other hand, with k_g increasing from zero, υ_{gr} first increases, reaches a maximum smaller than c, and then decreases until at k_g = π/d it reverts to υ_{gr} = 0.
The dispersion curve for a disk-loaded waveguide thus deviates from that of a uniform waveguide and, as shown in Fig. 13.13, exhibits discontinuities at k_g = nπ/d, with n an integer. The discontinuities in frequency ω separate regions of ω that can pass through the disk-loaded waveguide (pass bands) from regions of ω that cannot pass (stop bands). Two such bands are shown in Fig. 13.13: a pass band for frequencies ω between \left(ω_{c1}\right)\ and\ \left(ω_{c2}\right) in light grey color, and a stop band for frequencies between \left(ω_{c2}\right)\ and\ \left(ω_{c3}\right). The region between k = −π/d and k = +π/d is called the first Brillouin zone and energy E versus wave number k diagram is called the Brillouin diagram.
A closer look at the disk-loaded dispersion relationship curve of Fig. 13.13 shows the following features:

(1) For angular frequencies ω in the range \left(ω_{c1}\right) ≤ ω ≤ \left(ω_{c2}\right) in the first pass band (also called the first Brillouin zone) frequencies in the region between points C and F on the dispersion plot have a phase velocity υ_{ph} smaller than or equal to c as a result of α_{ph} ≤ arctan c. Loading the uniform waveguide with disks thus decreases the phase velocity below c for certain angular frequencies ω, opening the possibility for electron acceleration with radiofrequency microwaves.

(2) Frequency \left(ω_{c2}\right) clearly has a phase velocity υ_{ph} which is smaller than c; yet, the frequency \left(ω_{c2}\right) would not be suitable for electron acceleration despite υ_{ph}c because, simultaneously, at frequency \left(ω_{c2}\right), the group velocity of the wave is zero (υ_{gr} = 0; tangent to dispersion curve at point F is horizontal).

Since the velocity υ_{en} of energy flow in a waveguide equals to group velocity υ_{gr} of microwaves propagating in the waveguide, it is obvious that for υ_{gr} = υ_{en} = 0 energy transfer from the microwave to the accelerated electrons is not impossible and thus points such as point F will not be suitable for acceleration despite satisfying the necessary (but not sufficient) condition υ_{ph} < c.

(3) However, there are frequencies in the frequency pass band between \left(ω_{c1}\right)\ and\ \left(ω_{c2}\right), such as angular frequency ω for point D on the dispersion plot in Fig. 13.13, for which v_{\mathrm{ph}} \lesssim c \text { and at the same time } v_{\mathrm{gr}}>0, and these frequencies are suitable for electron acceleration.

(4) In practice, frequencies which give υ_{ph} smaller than yet close to c, i.e., v_{\mathrm{ph}} \lesssim c, are used for electron acceleration in disk-loaded waveguides. The group velocities for these frequencies between points C and D on the dispersion curve of Fig. 13.13 are non-zero but nonetheless very low, so that for a typical accelerating waveguide the phase velocity is about two orders of magnitude larger than the group velocity \left(υ_{ph}/υ_{gr} ≈ 100\right).