# Wide Band Interaction Impedance and Mode Excitation in Glide Symmetric
Double Corrugated Waveguides for mm-wave TWTs

Nelson Castro1, Miguel Saavedra-Melo2, Eva Rajo-Iglesias 1, Filippo Capolino21
1Department of Signal Theory and Communications, University Carlos III of
Madrid, Leganés, Spain  2Department of Electrical Engineering and Computer
Science, University of California, Irvine, California, USA

###### Abstract

Focusing on traveling wave tube (TWT) applications, the interaction impedance
between an electron beam and electromagnetic modes in three distinct, but
related, corrugated waveguides that operate at millimeter waves is
investigated together with the role of glide symmetry. Two waveguide
structures have glide symmetry, and the irreducible Brillouin zone is related
to half of the period, leading to a wide band linearity, i.e., nondispersive,
property of the dispersion diagram. The investigation of the modes with
longitudinal electric field that can be excited shows that the bottom-top
glide symmetric corrugated waveguide has a wide band interaction impedance,
hence it is a good candidate for millimeter wave TWT amplifiers.

##  I Introduction

Slow Wave Structures (SWS) are RF structures where the electromagnetic wave is
confined and the phase velocity is less than the speed of light. The two main
SWS-based devices are the traveling wave tube amplifier (TWT), where the wave
and power flow travel in the same direction, and the backward wave oscillator
(BWO), where the wave propagates in the opposite direction to which the energy
flows [1]. In a TWT, an input signal is amplified due to the synchronous
interaction between the slow wave within a periodic structure and an electron
beam along a narrow vacuum channel [2], in other words, as the beam and wave
go forward, the beam is modulated by the electric field of the wave and the
modulated beam increases the amplitude of the electric field [1].

Research on TWT has increased considering its ability to generate high power
at millimeter waves, which are useful in multiple applications [3, 4, 5, 6, 7,
8, 9]. Advantages of TWTs include moderate beam voltage, moderate structure
period to wavelength ratio, high beam power to radiated power transformation
efficiency [10], long lifetime [11], and reliability and robustness against
high thermal stresses and ionizing radiation typical of the space environment
in most space and airborne applications [12]. However, the most interesting
advantage is its very wide instantaneous bandwidth, which allows signals to be
amplified linearly down to the kilowatt level, even in the millimeter
wavelength (mmW) bands [7], where TWTs are excellent devices for communication
systems, especially when broadband transmission with moderate power
amplification is required in a compact size [13]. The latter makes them ideal
for applications such as communications, advanced radar, military electronic
countermeasure systems, and space research [5].

Standard TWTs are generally based on helical structures that can provide
multi-octave bandwidths and power outputs in excess of 100 W up to 50 GHz
[12]. However, since it is a very complex geometry, its fabrication is
particularly challenging as the frequency increases, which becomes impractical
for operating frequencies above 65 GHz [12]. In the millimeter wave and THz
bands, the most popular SWS is the serpentine waveguide which can provide wide
bandwidth and moderate output power, but as before, at very high frequencies
the complexity of the structure makes it a difficult task to perform [7]. In
recent years, straight rectangular SWSs have been introduced which are
relatively easy to fabricate since the beam tunnel may be easier to integrate
with the structure, as it can be seen in half-period staggered double vane
structures using sheet-beam current proposed for a 220 GHz TWT amplifier [14],
[15], and double corrugated rectangular SWSs to support circular electron
beams in the microwave regime and also feasible for the THz band [12, 16, 17,
18, 19].

Glide symmetry in periodic structures like the TWT was originally studied in
[20, 21, 22, 23] and has gained notable interest as this feature has several
advantages, such as low dispersion, wide stopbands, and high levels of on-axis
anisotropy (used in lens antenna design) [24]. Glide symmetry was used in
electron-beam devices for long time including the 1950s [25, 26, 27, 28, 29]
but without knowing the deep spectral properties later highlighted in [23].
Glide symmetry was first studied in 1964 [20] and is a specific case of higher
symmetries, created with translation of half of the geometrical period,
followed by a reflection in a so-called glide plane. This type of high
symmetry, as well as screw symmetry, has been used in microwave structures
such as filters, traveling wave tubes, and traveling-wave antennas [23].
Recently, interest in glide symmetry has been resurrected with new and
fascinating applications [24, 30], such as widening the bandwidth and
increasing the attenuation of holey structures at the stopband, useful for
producing cost-effective gap waveguides [31]. There, a glide-symmetric holey
electromagnetic bandgap (EBG) structure has been proposed to be used to
manufacture low-cost waveguide structures at high frequencies. Regarding
recent TWTs, straight rectangular SWSs with glide symmetric staggered vane
interaction structure have been previously studied (see [32] and references
therein), leading to, for example, a 100-W, 200-GHz high bandwidth amplifier.
Furthermore, an offset double corrugated SWS as enabling power amplifier for
long range sub-THz links was proposed in [33] and in [34] where the pillars
(i.e, corrugations) exhibit glide symmetry (a structure here referred to as
”BB Glide”, see Fig. 1), though no comprehensive data and results were shown.

In this work, the bandwidth enhancement as a result of the presence of glide
symmetry in corrugated waveguides is explored. The glide plane is allocated in
two different configurations and the interaction impedance and field
polarization at the first space harmonic are studied and compared to the
results pertaining to the double corrugated waveguide without glide symmetry
that has been previously published [12, 16, 17, 18], referred here as ”BB”
(see Fig. 1). By looking at the spatial $\beta$ spectrum, we also establish
which modes can be excited in the glide symmetric structures and lead to the
conclusion that the newly proposed structure with glide symmetry, here
referred to as ”BT Glide” (Fig. 1) leads to very wide band, dispersionless,
operation (Fig. 2).

##  II Glide Symmetry and Waveguide Geometries

(a)

(b)

(c)

Figure 1: Unit cells of the three SWSs supporting modes propagating in the $z$
direction, made of a metallic waveguide with periodic pillars (the corrugation
of period $d$) of square cross-section: (a) Bottom-bottom corrugations (BB);
(b) Bottom-bottom corrugations with glide symmetry (BB Glide); (c) Bottom-top
corrugations with glide symmetry (BT Glide).

###  II-A Glide symmetry

Glide symmetry is in the two waveguides whose unit cells are shown in Fig. 1
(b) and (c). A periodic structure is said to possess glide symmetry if it
remains invariant under the glide operation $G$ that consists of a translation
of half a period $d$, followed by a reflection [23]. The $G$ operation for the
BB Glide structure in Fig. 1(b) is defined in Cartesian coordinates as

$G=\begin{cases}x\rightarrow-x\\\ y\rightarrow y\\\ z\rightarrow
z+d/2\end{cases}$ (1)

whereas, for the BT Glide structure in Fig. 1(c), the $G$ operation is defined
as

$G=\begin{cases}x\rightarrow-x\\\ y\rightarrow-y\\\ z\rightarrow
z+d/2\end{cases}$ (2)

In any periodic structure without higher symmetry (like the BB case), the
modes exhibit band edges (between a pass and a stop band) at $\beta
d/\pi=-1,0,1,2,...$, where the group velocity vanishes. Something different
happens for the glide symmetry cases, like the BB Glide and BT Glide
geometries. When a structure has glide symmetry, the upper and lower branches
may merge, as it will be clear next, and the group velocity does not vanish at
the edge or at the center of the BZ: for example, band gaps disappear at
$\beta d/\pi=0,\pm 2,\pm 4,...$ for the cases in Sec. II-IV, whereas band gaps
disappear at $\beta d/\pi=\pm 1,\pm 3,...$ for the case studied in Appendix.
The fundamental reason is that, as explained in [23], the Bloch-Floquet
theorem can be applied to the fundamental cell of length $d/2$ when glide
symmetry is present, and not to the full period $d$. In other words, for glide
symmetric periodic waveguides the dispersion curves flatten with a spectral
period of $4\pi/d$, instead of the usual spectral period of $2\pi/d$ for non-
glide symmetric structures.

###  II-B Three slow Wave structures

The so-called double corrugated waveguide (DCW) [12] has been proposed as a
candidate for TWT amplifiers and BWOs [16]. One of its advantages is the
wideband modal interaction with the electron beam if the first higher space
harmonic is used (i.e., $n=+1$).

One desired feature of amplifiers is the large operational bandwidth. In order
to improve the wideband behavior of the DCW, the implementation of glide
symmetry into the structure is here explored. Three different topologies are
studied. First, the classic DCW, which is shown in Fig. 1(a), and will be
referred to as BB (Bottom - Bottom) in this paper. Then, glide symmetry is
implemented considering two different glide planes: first, applying glide
symmetry in the $x-z$ plane, as in the unit cell shown Fig.1(b) that will be
referred to as BB Glide, which corresponds to the offset(O)-DCW in [33]; and
finally glide symmetry is in both the $x-z$ and $y-z$ planes, as shown in Fig.
1(c) and referred to as BT (Bottom - Top) Glide. An interesting case of BB
Glide symmetry is further analyzed in the Appendix. The three structures will
be analyzed in terms of the dispersion diagram, where the mode excitation and
polarization of each space harmonic will be considered, and then later on
looking at the spectrum of the excited modes, and finally in terms of
interaction (Pierce) impedance.

(a)

(b)

(c)

Figure 2: Modal dispersion diagrams for the three SWS geometries. We include
also the polarization of the field associated with each Floquet harmonic. (a)
BB, (b) BB Glide, (c) BT Glide. Continuous lines denote harmonics with
positive group velocity, dashed lines denote the harmonics with negative group
velocity, and the beam line $\beta_{0}=\omega/u_{0}$ (purple) denotes the
electron beam’s charge wave with velocity $u_{0}=0.28c$ for the BB and BB
glide cases, and $u_{0}=0.275c$ for the BT glide case, where $c$ is the speed
of light. When using glide symmetry in (b) and (c), the two modal branches
(green and blue) in (a) merge and there are no more band edges at $\beta
d/\pi=\pm 1,\pm 3,...$. This phenomenon, induced by glide symmetry, provides a
much large bandwidth for synchronization with an electron beam.

###  II-C Dispersion diagrams

The dispersion diagram of the modes of each waveguide is computed using the
eigenmode solver of the commercial software, CST Microwave Studio assuming the
metallic waveguides are made of perfect electric conductor. The unit cells are
optimized to have the $n=+1$ harmonic of the EM mode interact (synchronize)
with an electron beam with a charges’ velocity of 0.28$c$ for the BB and BB
glide cases, and $u_{0}=0.275c$ for the BT glide case, being $c$ the speed of
light. This leads to the following dimensions (in mm), $a=2.7$, $b=2.1$,
$a_{pin}$ = 0.4, $h_{pin}$ = 0.9, $d=2.2$. For the BB and the BB Glide cases,
we assume $g=0.3$ mm, whereas for the BT Glide case we have $c=0.3$ mm and
$g=0$ mm. The values of $g$ and $c$ are equal for comparison purposes. (A
further case of BB glide symmetry is analyzed in the Appendix.)

The obtained dispersion diagrams are plotted in Fig. 2. For reference
purposes, the cutoff frequency of the straight waveguide without corrugations
is 55.5 GHz. The double corrugation reduces the cutoff frequency of the Bloch
mode in the periodic waveguide.

We use the common definition for the Brillouin zone (BZ), which implies that
the fundamental (0th order) BZ is defined for $-1<\beta d/\pi<1$. Therefore
the $n=1$ Floquet harmonic is defined for $1<\beta d/\pi<3$, denoted as first
high order BZ. In any periodic structure without higher symmetry (like the BB
case), the modes exhibit band edges (between a pass and a stop band) at $\beta
d/\pi=-1,0,1,2,...$, where the group velocity vanishes. Something different
happens for the glide symmetry cases, like the BB Glide and BT Glide cases:
the group velocity does not vanish anymore at the BZ edges $\beta d/\pi=\pm
1,\pm 3,...$, because the upper and lower modal branches that were visible in
Fig. 2(a) are now merged. This has a major implication since it is evident
that there are wide frequency bands that can be synchronized with the electron
beam, as apparent for the BB Glide and BT Glide cases.

For the BB case, the modal dispersion diagram in Fig. 2(a) is showing the
first two lower modal branches, where a stop band exists between them. The
interaction with an electron beam may occur between 45 GHz and 53 GHz, and a
fractional bandwidth of 16% is obtained at 50 GHz which is in line with
previously published works with this topology. When glide symmetry comes into
the structure, the stop band between the two modes is eliminated leading to a
larger bandwidth, for both BB and BT Glide SWSs because of the non dispersive
$k-\omega$ relation over a very wide frequency range, especially for the BT
Glide case. In the BB Glide case, the diagram in Fig. 2(b) shows that the
fractional bandwidth is around 28%, whereas for the BT Glide SWS, the diagram
in Fig. 2(c) shows a bandwidth of approximately 50%, i.e., between 40 GHz to
65 GHz.

It is necessary then to study the mode polarization associated to the branches
of the dispersion diagram; this information is important to understand what
happens when a source is located at one end of the waveguide, as the
intersection of the branches of the dispersion diagram at odd values of $n$
may lead to unwanted behavior.

The solid lines in the dispersion diagrams in Fig. 2 represent the excited
branches by a source at the left end of the waveguide, since these modes have
positive group velocity and carry energy along the $+z$ direction. The
polarization of each harmonic was determined by performing the Fourier
analysis of each component of the electric field phasor, $E_{x}(z)$,
$E_{y}(z)$ and $E_{z}(z)$, solution of the eigenmode solver. The results show
that for different geometries, different polarizations of the $nth$ field are
obtained. For the BB case, all branches have the same polarization but when
symmetry is implemented, the branch associated with the $n=\pm 1$ harmonic has
different polarization depending on which symmetry plane is taken into
account. It is known that in order to have synchronization with the electron
beam the field must be polarized in the direction in which the electrons are
traveling [1]; from this analysis, then, is manifested that the BB glide is
not suitable for wideband interaction at the first space harmonic (i.e.,
$n=1$), this will be confirmed later on when the interaction impedance for
this structure is computed. Instead, for the BT Glide case, the field is
indeed polarized in the wanted direction since the $n=1$ harmonic is mainly
polarized along $z$.

##  III Mode and Spatial Harmonic Excitation

In the previous section, the polarization associated with each branch of the
dispersion diagram was shown based on the eigenmode solver calculations. In
this section, we investigate what modes are actually excited and what are the
polarizations of the spatial harmonics when a finite-length SWS is excited by
Port 1 on the left, as shown in Fig. 3. The simulation setup utilized to
obtain the phasor of the electric field to perform the spatial Fourier
transform in the $z$ direction is presented in Fig. 3. Time-domain full-wave
simulations have been performed for a waveguide of length $l=20d$, where
$d=2.2$ mm is the period of the unit cell and the structure is excited using
the waveguide Port 1. The field is calculated along a line in the $z$
direction, at position $\delta$, for the BB and BB Glide $\delta=0.9$ mm and
for BT Glide $\delta=b/2=1.05$ mm. For each case, we choose waveguide ports as
follows. For the BB case, Port 1 excitation corresponds to a conventional
waveguide $TE_{10}$ mode as can be seen in Fig. 3(a) and to better match this
port mode to the mode of the corrugated waveguide with lower cutoff frequency,
an extra waveguide length between the port and the pillar is added. For the BB
Glide and BT Glide cases, the port is located in contact with a pillar; in
both cases, the solver’s port analysis produces the excitation using a mode
transverse profile that is similar to that of the Bloch mode in the corrugated
waveguide.

(a)

(b)

(c)

Figure 3: Simulation setup of the three corrugated waveguides with the
corresponding mode excited at the left port: (a) BB case where the light blue
waveguide is a small extension used in the computation to have the port
slightly away from the pillars; (b) BB Glide; (c) BT Glide. In these latter
two cases, the port includes the section of a metallic pillar.

(a)

(b)

(c)

Figure 4: Spatial spectrum of the electric field components $E_{x}$, $E_{y}$,
and $E_{z}$ at 50 GHz excited by a source at the left end of the SWS: (a) BB
case; (b) BB Glide case; (c) BT Glide case. We are interested in generating a
strong $E_{z}$ component in the $n=1$ space harmonic. The red dots represent
the amplitudes related to the branches with positive group velocity, and the
peaks with no dots are related to the waves reflected toward the source from a
SWS truncation at the right end.

The phasor of each electric field component is numerically calculated on a
scan line along the $z$ direction (red line) and exported to perform the
Fourier transform of the excited field in the periodic waveguide

$\displaystyle\tilde{E}_{v}(\beta)=\int_{z=0}^{z=l}E_{v}(z)e^{j\beta z}dz.$
(3)

Here, $\beta$ is the spectral wavenumber and $E_{v}(z)$ is the polarization
component of the electric field, with $v=x,y,z$. The spatial spectrum
$\tilde{E}_{v}(\beta)$ of the excited field in each waveguide at 50 GHz is
shown in Fig. 4, as $20\log\left[|\tilde{E}_{v}/l|/(1\rm{V/m})\right]$. The
amplitudes of the red-dotted peaks are related to the mode with positive group
velocity, at different spatial harmonics $n$. In all cases, the frequency of
the observed peaks is in agreement with the dispersion diagram at 50 GHz shown
in Fig. 2.

In the BB case, we observe a $z$ polarized wave in the $n=-1$, $n=0$, and
$n=1$ harmonics.

For the BB Glide in Fig. 4(b), in the fundamental Brillouin zone region from
$-1<\beta d/\pi<1$ the field only has components related to $y$ and $z$
polarizations, but when looking at the first harmonic region, $n=1$, the field
component is $x$ polarized, and there is no $z$ polarized field. Furthermore,
in the fundamental Brillouin zone the BB glide structure exhibits a fast-wave
behavior since the phase velocity is greater than the speed of light in most
of the frequency band, which prevents synchronization with the electron beam.

On the other hand, for the BT Glide case, the peaks of the amplitude of the
field are $y$ and $x$ polarized for the fundamental BZ (i.e., $n=0$) and $z$
polarized for the harmonic $n=1$. Therefore, the BT Glide SWS may be chosen as
a SWS for a TWT operating with the $n=+1$ spectral region.

This analysis allows us to evaluate which SWS is suitable for a TWT amplifier
application since, as stated above, to synchronize a linear electron beam with
the RF mode we need a strong $E_{z,n}$ in the spectral region $n=1$. We
conclude that if wide band interaction is required, the BB Glide SWS cannot be
used to work in the $n=+1$ spectral region ($E_{z,1}$ is negligible in Fig.
4); however this structure could be used in a TWT if the RF wave is slow
enough and $d<<\lambda$ using a large interaction impedance at the fundamental
harmonic since $E_{z,0}$ is strong. For the BT Glide case, it is possible to
achieve a very wide band synchronization at the first ($n=1$) harmonic region
to design wideband power amplifiers.

An important observation from the results of this section is that a Port 1
excitation would not excite the mode with negative group velocity shown in
Fig. 2 with dashed curves. A Port 1 excitation at 50 GHz would excite only the
mode with positive group velocity. Modes associated with dashed curves can be
excited only by a wave reflected by Port 2. However, the higher order mode for
the BB case (green line, in Fig. 2)) could synchronize with the electron beam
generating self-sustained oscillations. The higher mode for the BB case (green
curve, in Fig. 2) is not present anymore for the BB Glide and BT Glide cases
because it merged with the lower mode (Fig. 2), thanks to glide symmetry; as
demonstrated in Fig. 4 only the positive-group velocity mode (solid blue, in
Fig. 2) can be excited by Port 1. The negative-group velocity branches (dashed
green, in Fig. 2) are not excited by Port 1. Therefore in a TWT device, the
electron beam would not synchronize with a backward EM wave, eliminating the
distributed feedback mechanism option for a BWO. One of the most important
observations for a TWT application is that in the BB Glide and even more for
the BT Glide cases, the BW of synchronization is very broad, as expected since
the two SWSs are glide symmetric.

##  IV Interaction impedance

In a TWT, the coupling between the RF wave and the electron beam is described
by the interaction (or Pierce) impedance, $Z_{n}$, which is the ability of
said wave to exchange energy with the synchronized electron beam. A large
interaction impedance represents the ability of the SWS to guide modes with a
strong $E_{z}$ field at the location where the electron beam would flow [2].
The interaction impedance is defined as a measure of the longitudinal electric
field acting on the charges for a given total power flow in the structure [1],
that is

$Z_{n}=\frac{|E_{z,n}|^{2}}{2\beta_{n}^{2}P}\qquad n=0,\pm 1,\pm 2,...$ (4)

where $\beta_{n}=\beta_{0}+2\pi n/d$, is the wavenumber of the $n$th Floquet
harmonic, $P$ is the mode time-average electromagnetic power flow [2].
Finally, $E_{z,n}$ is the phasor representing the longitudinal component of
the $n$th harmonic of the electric field $E_{z}(z)$ in the periodic SWS,
obtained from the Floquet-Bloch series expansion
$E_{z}(z)=\sum_{n=-\infty}^{\infty}E_{z,n}e^{-j\beta_{n}z}$. The harmonic
strength $E_{z,n}$ is found as

$E_{z,n}=\frac{1}{d}\int_{0}^{d}E_{z}(z)e^{j\beta_{n}z}dz.$ (5)

Figure 5: Interaction impedance for the $n=+1$ Floquet harmonic with positive
group velocity, in the three SWSs under study. The BT Glide SWS exhibits a
very large operating frequency band because of glide symmetry. The BB Glide
case has a vanishing $E_{z}$ component in the $n=1$ harmonic and therefore it
shows a vanishing impedance. The BB case has a narrower bandwidth because the
structure does not have glide symmetry.

(a)

(b)

(c)

Figure 6: Parametric study of the interaction impedance $Z_{1}$, associated to
the dispersion branch with positive slope, for different spacings between the
pillars (i.e., corrugations) for the BT Glide case: (a) Different values of
$c$ while $g=0$ mm. (b) Different values of separation $g$ while $c=0$ mm; the
impedance values are comparable with those of the BB geometry but BT Glide
shows much wider bandwidth. (c) Different values of separation with increments
in $c$ and $g$.

In this paper we are particularly interested in the branch with positive slope
of the $n=+1$ harmonic, which is the one that would interact with the electron
beam, aiming for wideband velocity synchronization, and where the group
velocity is positive and the SWS can be used as an amplifier. The interaction
impedance versus frequency for the positive-slope branch of the modes in the
three geometries is plotted in Fig. 5. We can see that, as expected, for the
BB and BT Glide cases we have a nonzero interaction impedance, but for the BB
Glide case, the impedance is vanishing because of the null of the longitudinal
$E$-field component of the $n=+1$ harmonic (see also the spectrum in Fig. 4)
as observed also in [33] for the $n=1$ harmonic component of the lower-band
mode (see also the distinct case in Appendix where a more complex mode
structure exhibits a non-vanishing impedance for the upper mode with positive
slope).

In the BB and in the BT Glide cases, the results show that $Z_{n}\geq 4$
$\Omega$ over the relative frequency bands; this value can be optimized by
changing the dimensions of the corrugated waveguide such as pins height, beam
position, beam channel width, and so on. What is very clear is that the use of
glide symmetry dramatically increases the bandwidth to more-or-less a relative
value of 34% without causing any reduction in the interaction impedance
compared to the BB case.

Additionally, a parametric study on the relationship between the interaction
impedance and the spacing available in between corrugations to fit the
electron beam is performed. The variable parameters are $c$ and $g$, shown in
Fig. 1. First, in the original BT Glide case, where $g=0$ mm, the spacing is
controlled by $c$, leading to the results in Fig. 6(a), with a range from
$c=0.3$ mm to $c=0.6$ mm, with a step of $0.15$ mm. A similar procedure is
performed for the results in Fig. 6(b) by setting $c=0$ mm so that the spacing
is controlled by $g$ only, and the results are compared with those of the BB
case because in such a case the distance between pillars depends exclusively
on $g$. Finally, for the results in Fig. 6(c), both $c$ and $g$ are set to
have a spacing from $0.3$ mm to $0.6$ mm in the cross-sectional diagonal
between the corrugation pillars.

From the latter results, we can see that the interaction impedance decreases
as the separation between pillars increases, with the worst case being
$Z_{n}\approx 1$ $\Omega$ when the distance is $0.6$ mm. However, the
comparison shown in Fig. 6(b) shows that the decreasing with the interaction
impedance with increasing spacing between corrugations is comparable with that
of the BB geometry, and that the wideband benefits of glide symmetry are still
present: this means that BB and BT Glide will have a similar (or equal) value
of impedance, but in presence of glide symmetry the bandwidth obtained is much
larger. The precise selection of the geometric values will depend on the beam
velocity, operating frequency band, required beam current, etc; the BT Glide
and BB geometries offer several degrees of freedom in the design, including
the period $d$ to make the dispersion diagram overlappoing the beam line over
a wide frequency band.

##  V Conclusion

We have presented a study of the interaction impedance and mode excitation in
three double corrugated rectangular waveguides, where one of these, the BT
glide structure, is a novel structure that uses glide symmetry in both the
$x-z$ and the $y-z$ planes. The presented structure is suitable to operate
around 50 GHz in the first (i.e., $n=1$) space harmonic, where the interaction
impedance is greater than 4 Ohm while the fractional bandwidth is
approximately 50 %, which is significantly greater than the bandwidth
exhibited by structures without glide symmetry or where the glide symmetry is
only applied over one plane. Furthermore, it was shown that depending on the
plane in which the glide operator is applied, the excited mode can have a
stronger or weaker longitudinal component. This is critical if wideband
interaction is required at a given Floquet harmonic.

Additionally, a parametric study of the interaction impedance as a function of
the distance between corrugations (i.e., pillars) was performed, leading to
the conclusion that glide symmetry leads to an improvement in terms of
bandwidth without any reduction of the interaction impedance compared to the
related non-glide BB corrugated waveguide.

## Acknowledgment

This material is based upon work partly supported by the Air Force Office of
Scientific Research Multidisciplinary Research Program of the University
Research Initiative (MURI) under grant number FA9550-20-1-0409 administered
through the University of New Mexico. The work was partially funded by the
Spanish Government under the grant PID2019-107688RB-C21 from
MCIN/AEI/10.13039/501100011033. FC Thanks the Universidad Carlos III de Madrid
for receiving a Chair of Excellence that made this research and collaboration
possible. The authors thank DS SIMULIA for providing CST Studio Suite.

## A different BB Glide SWS

In some cases, the BB Glide geometry in Fig. 1(b) leads to interesting
dispersion diagrams and useful values of the interaction impedance associated
to the $n=1$ Floquet harmonic modal branch with positive group velocity. Let
us consider for example a DCW with BB Glide symmetry as in Fig. 1(b), with
unit cell dimensions (in mm): $a=1.8$, $b=0.45$, $a_{pin}$ = 0.09, $h_{pin}$ =
0.327, $d$ = 0.505 and $g$ = 0.17. Some of these dimensions are the same as
those in [33]. The cutoff frequency of the first mode in the waveguide without
corrugations is $f=83.3$ GHz.

The dispersion diagram of the modes in the waveguide is shown in Fig. 7(a). It
is rather different from what is presented in Section II: There are two
branches with positive group velocity in the region $2<\beta d/\pi<3$, the
solid green and the solid blue (see Fig. 2(b) for comparison). The upper mode
in solid green has a positive group velocity in the region $2<\beta
d/\pi<2.8$, meaning that this modal branch can be used for synchronization
with the electron beam to design a TWT amplifier. The interaction impedance is
calculated using (4) and plotted as a function of frequency in Fig. 7(b), for
both the solid-blue lower and solid-green upper modes: the solid blue mode has
a vanishing interaction impedance, exhibiting a behavior similar to the BB
Glide case shown in Fig. 5. Instead, the interaction impedance plotted in the
solid green related to the upper solid-green branch in Fig. 7(a), shows that
this mode is suitable for the interaction, having a non-zero interaction
impedance (an analogous conclusion was reached in [33, 34]). However, even
though the interaction impedance is non vanishing for the $n=1$ harmonic
component of the higher-band mode, the extended bandwidth as a consequence of
the glide symmetry is not present. Indeed, interesting observations are made
when we compare the dispersion diagrams in Fig. 2(b) and in Fig. 7(a), both
relative to BB Glide SWSs.

The dispersion diagram in Fig. 2(b) shows: (i) a very wide band resulting from
the disappearing of the bandgap at $\beta d/\pi=3$, which is in the middle of
the branch of positive slope, and (ii) there are no other modes in the same
frequency range, besides the presence of the mode with negative slope.

The dispersion diagram of the solid-green upper mode in Fig. 7(a) shows: (i) a
much smaller bandwidth, despite the disappearing of the bandgap at $\beta
d/\pi=3$; (ii) the solid-blue lower mode with positive slope over a wide
frequency band, which has a vanishing interaction impedance, still shows the
disappearing of the bandgap at $\beta d/\pi=3$, because of glide symmetry;
(iii) both modes however experience the closure of the bandgap at $\beta
d/\pi=3$ close to their upper band edge; (iv) there are two modes with
positive slope, besides the two modes with negative slope. In other words, the
dispersion diagram shown in Fig. 7(a) is complex and it is the result of few
mode interactions. One can also recognize that both dispersion curves have a
minimum (either at $\beta d/\pi=0$ or at $\beta d/\pi=2$) between two maxima,
which is a condition related to the split band edge (SBE) discussed in [35,
36] and that may lead also to the presence of a degenerate band edge (DBE)
[35, 37, 38] when properly selecting the dimensions. A proposed application of
the DBE in linear beam devices is the ”multimode synchronization regime” for
low threshold BWOs [39, 40].

(a)

(b)

Figure 7:  Results of the BB Glide DCW with dimensions slightly different from
the ones in Secs.II-IV.(a) Dispersion diagram, with the beam line plotted for
a voltage of 13.5 kV (0.23$c$). The interaction is achieve for the mode in
solid green between 150 GHz and 165 GHz. (b) Interaction Impedance for the two
$n=1$ modal branches with positive slope, in the region $2<\beta d/\pi<3$.

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