# Three-Way Serpentine Slow-Wave Structures with Stationary Inflection Point
and Enhanced Interaction Impedance

Robert Marosi, Tarek Mealy Alexander Figotin, and Filippo Capolino Robert
Marosi, Tarek Mealy, and Filippo Capolino are with the Department of
Electrical Engineering and Computer Science, University of California, Irvine,
Irvine, California, e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Figotin is with the Department of Mathematics,
University of California, Irvine, Irvine, California, e-mail:
<EMAIL_ADDRESS>
###### Abstract

We introduce two novel variants of the serpentine waveguide slow-wave
structure (SWS), often utilized in millimeter-wave traveling-wave tubes
(TWTs), with an enhanced interaction impedance. Using dispersion engineering
in conjunction with transfer matrix methods, we tune the guided wavenumber
dispersion relation to exhibit stationary inflection points (SIPs), and also
non-stationary, or “tilted” inflection points (TIPs), within the dominant
$\mathrm{TE_{10}}$ mode of a rectangular waveguide. The degeneracy is found
below the first upper band-edge associated with the bandgap where neighboring
spatial harmonics meet in the dispersion of the serpentine waveguide (SWG)
which is threaded by a beam tunnel.

The structure geometries are optimized to be able to achieve an SIP which
allows for three-mode synchronism with an electron beam over a specified
wavenumber interval in the desired Brillouin zone. Full-wave simulations are
used to obtain and verify the existence of the SIP in the three-way coupled
waveguide and fine-tune the geometry such that a beam would be in synchronism
at or near the SIP. The three-way waveguide SWS exhibits a moderately high
Pierce impedance in the vicinity of a nearly-stationary inflection point,
making the SWS geometry potentially useful for improving the power gain and
basic extraction efficiency of millimeter-wave TWTs. Additionally, the
introduced SWS geometries have directional coupler-like behavior, which
enables distributed power extraction at frequencies near the SIP frequency.

###### Index Terms:

serpentine waveguide (SWG), serpentine ladder waveguide (SLWG), three-coupled
serpentine waveguide (TCSWG), traveling-wave tube (TWT), millimeter-wave,
dispersion, stationary inflection point (SIP), distributed power extraction
(DPE), interaction impedance.

##  I Introduction

Traveling-wave tubes (TWTs) are able to perform broadband, high-power
amplification due to the distributed transfer of energy from a beam of
electrons to guided electromagnetic fields. There are two primary factors that
control the strength of the interaction between the beam and the
electromagnetic field: velocity synchronization between the beamline and
guided modes and beam-wave interaction impedance, also called Pierce
impedance, in the TWT slow wave structure (SWS) [1]. In S-band helix TWTs,
Pierce impedance is typically on the order of 100 Ω, allowing for efficient
basic beam-wave power conversion. However, at millimeter-wave frequencies,
microfabricated slow-wave structures such as the serpentine waveguide (SWG)
typically exhibit Pierce (or interaction) impedance on the order of 10 Ω or
smaller, which drastically reduces their basic conversion efficiency. This is
partly due to the fact that serpentine-type TWTs are typically synchronized to
an electron beam in the first or second Brillouin zone rather than the
fundamental Brillouin zone to avoid the use of relativistic electron beam
velocities which require much larger voltages to accelerate electrons close to
the speed of light [2]. Due to structure geometry that scales inversely with
operating frequency, helix-type TWTs become extremely difficult to fabricate
at millimeter-wave frequencies, making microfabricated structures like the SWG
much more attractive. In this paper, we propose two new types of dispersion-
engineered SWSs based on the SWG geometry that are capable of exhibiting
moderately high Pierce impedance over narrow bandwidths and can be
microfabricated. These two structure variants, the serpentine ladder waveguide
(SLWG) and the three-coupled serpentine waveguide (TCSWG), with their
longitudinal cross-sections illustrated in Figs. 1a and 1b, respectively, can
have their geometries easily designed to exhibit stationary inflection points
(SIPs) or nearly-stationary tilted inflection points (TIPs), sometimes
referred to as tilted SIPs, at specific frequencies, similarly to the kind
shown in Fig. 1c. Such structures with SIPs are capable of exhibiting
moderately high to very high Pierce (interaction) impedance comparable to the
impedance observed near the band-edge of SWG SWS, where the group velocity and
power flow also vanish. Therefore, at the frequencies that the electron beam
is in synchronism with the SIP, we expect the proposed SIP SWS to also have a
high Pierce gain parameter, as $C^{3}=Z_{\mathrm{Pierce}}I_{0}/(4V_{0})$,
where $I_{0}$ and $V_{0}$ are the average current and equivalent kinetic
voltage of the electron beam, respectively. These structures have potential
use in the design of compact, high efficiency, millimeter-wave TWTs and
backward-wave oscillators (BWOs).

The SIP is a class of modal degeneracy, whereby three eigenmodes coalesce in
both their wavenumbers and eigenvectors (polarization states). Such modal
degeneracies of orders 2, 3, and 4 were originally investigated by Figotin and
Vitebskiy in [3, 4, 5, 6, 7, 8]. The SIP is a particular type of exceptional
point of degeneracy (EPD), and is sometimes called a “frozen mode” in
literature. Exceptional points of degeneracy of various orders have been
previously investigated theoretically in gainless and lossless structures
operating at both radio and optical frequencies in [9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22], and have also been experimentally demonstrated at
radio frequency in [23, 24, 25, 26, 27, 28]. In particular, the first
experimental demonstration of SIPs at radio frequencies in a reciprocal three-
way waveguide SWS has been performed in [28]. The slow wave structures we
introduce here are designed to operate with an electron beam synchronized to
three degenerate “cold” eigenmodes. That is, the SIP is made to exist in the
“cold” dispersion relation, i.e.,before the introduction of an electron beam,
which will perturb the dispersion relation. Note that this regime of operation
is different from the regime of “exceptional synchronization” or “degenerate
synchronization” studied in [29, 30, 31], where an exceptional point of
degeneracy is designed to occur in the “hot” system, i.e., it is visible only
in the hot modal dispersion relation. These exceptionally synchronized modes
of the structure become degenerate when a synchronized electron beam is
coupled to the electromagnetic modes.

We define a “waveguide way” as an individual waveguide component (rectangular
waveguide, SWG, etc.) which is not in cutoff over the designed operating
frequency and can support two electromagnetic modes (one forward and one
backward). A three-way waveguide supports six modes, when considering
propagation in both the $+z$ and $-z$ directions, i.e., three modes in each
direction. The three-way microstrip structure with SIP in [28] is what
inspired the design of the structures in this paper.

The concept of “three-mode synchronization regime” using SIPs in the cold SWS
dispersion of linear-beam tubes was initially proposed in [22] and was based
on ideal transmission lines, following a multi-transmission line
generalization of the Pierce model. Here, we show for the first time how one
can design a realistic serpentine-like SWS to exhibit a cold SIP at
millimeter-wave frequencies.

In Section II, we explain the concept of the SIP, smooth-TIP, and alternating-
TIP. In Section III, we introduce the two proposed waveguide structures and we
describe our design methodology to introduce SIPs in the three-way coupled
waveguides. In Section IV, we show the dispersion, scattering parameters, and
Pierce (interaction) impedance enhancement for our three-way coupled
waveguides. All dimensions used in this paper are in SI units unless otherwise
stated.

(a)

(b)

(c)

Figure 1: Longitudinal cross-sections in the _y-z_ plane showing half of the
periodic metallic structures composed by cascading (a) SLWG unit cells, and
(b) TCSWG unit cells with periodicity $d$. Electron beams (red) are
illustrated in beam tunnels. (c) Example of the dispersion relation of the
mode with purely real wavenumber in the infinite periodic structure where the
SIP occurs (black solid line) at a frequency $f_{\mathrm{SIP}}$ and
fundamental wavenumber $k_{\mathrm{SIP}}$ where three modes coalesce (for
clarity, only one mode is shown, the one with purely real wavenumber). The
dispersion relation in the vicinity of an SIP is cubic. The three-way
waveguide dimensions may also be slightly altered to make the SIP either a
smooth-TIP (blue dash-dotted line) or alternating-TIP (magenta dotted line),
rather than an SIP.

##  II Cold Stationary Inflection Points

Stationary inflection points are a special case of eigenmode degeneracy,
whereby three eigenmodes coalesce at a single frequency point in the
dispersion diagram for the modes of a periodic structure, as has been explored
in [32, 33, 34, 35, 10, 36, 37, 38, 39, 22, 12, 40]. On the other hand, a non-
stationary, or “tilted” inflection point is a single frequency point in the
structure’s modal dispersion diagram where three eigenmodes are nearly
coalescing, but not perfectly so. In the structure’s modal dispersion diagram,
the dispersion relation local to an SIP or TIP will be cubic in shape. In
general, these cubic-shaped dispersion relations will have an inflection point
that occurs at a frequency-wavenumber combination where the second derivative
of the $\omega-k$ dispersion relation vanishes (i.e. $d^{2}\omega/dk^{2}=0$).
These inflection points are classified here into two kinds: the SIP, and the
non-stationary inflection point, or TIP. The SIP occurs where both the first
and second derivative of the $\omega-k$ dispersion relation vanish at the same
wavenumber, i.e., $d\omega/dk=0$ and $d^{2}\omega/dk^{2}=0$. For the case of
the TIP, only the second derivative of the $\omega-k$ dispersion relation
vanishes at the inflection point. In other words, the SIP is a special case of
TIP. Because the modal dispersion diagrams for our lossless, reciprocal
structures are symmetric in each Brillouin zone, the usual classification of
non-stationary inflection points as rising or falling is ambiguous. Every
reciprocal waveguide with a TIP in its dispersion relation will always exhibit
two kinds of TIPs, rising and falling, for opposite signs of
$k_{\mathrm{TIP}}$. To remedy this, we futher classify TIPs into two sub-
categories: smooth-TIPs which have a non-vanishing group velocity that does
not change sign for wavenumbers slightly above and below the inflection point,
and alternating-TIPs which have a group velocity that alternates in sign as
the wavenumber is swept near the inflection point, as illustrated in Fig. 1c.

We were able to design both the SLWG and TCSWG structures to exhibit SIPs in
their cold dispersion, that is, without an electron beam present. We stress
that this form of modal degeneracy is different from the hot eigenmode
degeneracy, or “exceptional synchronization” studied in [29, 30]. The
coalescence of the cold eigenmodes at an SIP results in a perfect cubic
dispersion relation local to the SIP in the dispersion diagram,

$\left(f-f_{\mathrm{SIP}}\right)\simeq h\left(k-k_{\mathrm{SIP}}\right)^{3}.$
(1)

When the cold eigenmodes of a structure are nearly-coalescing at a single
wavenumber, and the dispersion relation exhibits a TIP, the dispersion
relation local to the inflection point may be represented by a depressed cubic
function (the quadratic term is suppressed due to the shift in $k$ by the
cubic function’s inflection point),

$\left(f-f_{\mathrm{TIP}}\right)\simeq
h\left(k-k_{\mathrm{TIP}}\right)^{3}+s\left(k-k_{\mathrm{TIP}}\right).$ (2)

The parameters $f_{\mathrm{SIP}}\simeq f_{\mathrm{TIP}}$ and
$k_{\mathrm{SIP}}\simeq k_{\mathrm{TIP}}$ are the frequency and Floquet-Bloch
wavenumber, respectively, at which the three eigenmodes coalesce or are
nearly-coalescing to form an SIP or a TIP, respectively. We also note that,
due to Floquet-Bloch spatial harmonics, the wavenumbers $k$,
$k_{\mathrm{TIP}}$, and $k_{\mathrm{SIP}}$ in (1) and (2) have a periodicity
of $2\pi/d$, where the pitch of the unit cell is $d$. That is, $k$,
$k_{\mathrm{TIP}}$, and $k_{\mathrm{SIP}}$ in these formulae do not
necessarily need to be within the fundamental Brillouin zone. The parameter
$h$ is a scalar flatness coefficient that depends on the strength of eigenmode
coupling and $s$ is a scalar coefficient that affects the “tilt” of the TIP,
as demonstrated in Fig. 1c. Tilted inflection points and their properties,
such as improved gain-bandwidth products and power efficiency, have also been
explored in [22], using a multi-transmission line generalization of the Pierce
model [41, 42].

At an SIP, the dispersion relation local to the inflection point is a cubic
function similar to Eqn. (2) with $s=0$, as shown in solid black in Fig. 1c.
At, and very close to, an SIP or smooth-TIP, the eigenwaves all propagate in
the same direction. That is, the group velocities of the eigenwaves do not
change sign at frequencies slightly lower and higher than $f_{\mathrm{TIP}}$.
At the inflection point of an SIP, the group velocity becomes zero. In a
smooth-TIP with $s>0$, the group velocity ($d\omega/dk$) at the inflection
point is no longer zero (blue dash-dotted line in Fig. 1c). Having the
electron beam interact with an TIP with a near-zero positive group velocity
(nearly-stationary TIP) instead of a perfectly-zero group velocity SIP may be
preferable for TWT designs, since the Pierce (interaction) impedance at the
frequency of the inflection point will be sufficiently large but the device
will not become absolutely unstable when the electron beam is introduced. On
the other hand, having an electron beam interact with a TIP that has negative
group velocity is useful for the design of BWOs. We call this interaction
between an electron beam and a cold SIP, a “three-mode synchronization” (see
[22] for more details). Beamline interactions at points of zero group
velocity, like the band-edge [43, 44, 45, 46] or the degenerate band-edge
(DBE) [17, 47], are to be avoided in the design of TWT amplifiers, as they are
considered a source of instability. The alternating-TIP (magenta dotted line
in Fig. 1c) is the second kind of TIP studied with $s<0$, in which the group
velocities of the eigenwaves will change sign at frequencies slightly lower
and higher than $f_{\mathrm{TIP}}$. If the geometry of a structure can be
tuned to exhibit smooth-TIPs for one set of dimensions and alternating-TIPs
for another set of dimensions, it is expected that such a structure can be
made to exhibit an SIP.

From Pierce theory, the Pierce (interaction) impedance for a specific Floquet-
Bloch spatial harmonic $p$ and specific wavenumber corresponding to the
frequency of interest is defined as

$Z_{\mathrm{Pierce}}(k_{p})=\frac{\left|E_{z,p}(k)\right|^{2}}{2\mathrm{Re}(k_{p})^{2}P(k)}$
(3)

where, $k_{p}=k+2\pi p/d$ is the wavenumber corresponding to the appropriate
pth Floquet-Bloch spatial harmonic, $p=0,\pm 1,\pm 2,...$, and the wavenumber
$k$ is in the fundamental Brillouin zone defined here as $kd/\pi\in[-1,1]$,
i.e., with $p=0$. Furthermore, $\left|E_{z,p}(k)\right|$ is the magnitude of
the phasor of the electric field component along the center of the beam tunnel
in the z direction for a given wavenumber and pth Floquet-Bloch spatial
harmonic, and $P(k)$ is the time-average power flux at the fundamental
wavenumber corresponding to the frequency of interest (the time average power
flux is the sum of power contributions from all spatial harmonics) [48]. To
obtain the magnitude of the axial electric field phasor, corresponding to the
appropriate spatial harmonic, the complex axial electric field along the beam
tunnel axis is decomposed into Floquet-Bloch spatial harmonics as
$E_{z}(z,k)=\sum_{p=-\infty}^{\infty}E_{z,p}(k)e^{-jk_{p}z}$, where the
harmonic weights are computed as
$E_{z,p}(k)=\frac{1}{d}\intop_{0}^{d}E_{z}(z,k)e^{jk_{p}z}dz$ [49]. Both the
complex axial electric field $E_{z}(z,k)$ and the time-average power flux
$P(k)$ through the cross section of the unit cell are calculated for the cold
structure (i.e. without the electron beam) using the eigenmode solver in CST
Studio Suite. However, one must pay careful attention to how the phase across
the periodic boundaries is defined in the CST model to correctly compute the
interaction impedance. Since the $\exp(j\omega t)$ time convention is used by
CST, the formula for calculating $E_{z,p}(k)$ requires a delaying phase from
the lower periodic boundary to the upper periodic boundary of the simulated
unit cell, i.e. phase of $E_{z}(z,k)$ must decrease from $z_{min}$ to
$z_{max}=z_{min}+d$ for a positive value of $k$. Conveniently, the
electromagnetic energy simulated within the enclosed vacuum space of the unit
cell between periodic boundaries in the eigenmode solver, which is based on
the finite element method (FEM) implemented in the software CST Studio Suite,
is always assumed to be 1 Joule for each eigenmode solution. Therefore, the
power flux is calculated using the formula $P=(1\,\mathrm{Joule})v_{g}/d$,
where $d$ is the unit cell pitch, and the group velocity $v_{g}=d\omega/dk$ is
determined directly from the dispersion diagram via numerical differentiation
(The group velocity is the same at every spatial harmonic).

In order for interaction impedance to be large , the ratio in Eqn. (3),
$\left|E_{z,p}(k)\right|^{2}/P(k)$, must become large in magnitude or the
wavenumber in the denominator must become very small (i.e., operating closer
to the fundamental spatial harmonic). At a nearly-stationary TIP, which is
close to becoming an SIP, the power flow at the inflection point is indeed
smaller than the power flow of conventional SWG at the same wavenumber of the
inflection point, this because the power flow is proportional to the group
velocity. Assuming that the magnitude of the axial electric field component is
comparable for both cases, one concludes that the Pierce impedance will be
larger for the structure with an inflection point than in a conventional SWG,
at the wavenumber corresponding to the inflection point.

Due to this phenomena, it is possible to obtain a moderately high, narrowband
Pierce impedance at an SIP or nearly-stationary TIP, which is several times
larger than the Pierce impedance observed in a conventional serpentine
waveguide, as demonstrated in Sec. IV.

Conventional TWT SWS exhibit higher symmetries, such as glide symmetry in the
serpentine-type TWT or screw symmetry in the helix-type TWT [50, 51]. Nearly
parallel dispersion curves are formed by breaking glide symmetry. Glide
symmetry can be broken in our structures by introducing minor dimensional
differences between two similar waveguide sections. This allows us to readily
tune the tilt of TIPs in our dispersion relation by simply varying one or more
of our structure’s dimensions.

Next, we show the design methodology for two kinds of SWS geometries, whose
unit cells are shown in in Fig. 2, that can be dispersion engineered to
exhibit SIPs or nearly-stationary TIPs.

##  III Proposed Waveguides exhibiting SIP

(a)

(b)

Figure 2: The two unit cell geometries of the two three-way waveguides studied
in this paper. The unit cells are constructed by periodically coupling three
waveguides. Only the vacuum space inside the waveguides is shown (i.e., no
metallic walls are shown). (a) SLWG unit cell with one square beam tunnel
threaded through the middle SWG. The middle SWG is coupled to two lateral
straight waveguides with larger waveguide heights ($b$ dimension) than the
middle SWG, (b) Dual-beam TCSWG unit cell with square beam tunnels threaded
through top and bottom SWGs. Background material is PEC. Waveguide ways have
subscripts _t_ ,_m_ , and _b_ (top, middle, and bottom) based on their
relative positions along the y-axis. Rectangular coupling slots which are in
cutoff are added between middle SWG bends and adjacent waveguide ways.

###  III-A Serpentine Ladder Waveguide (SLWG)

The SLWG was our first attempt at obtaining a SWG-like structure which is
capable of exhibiting an SIP and has its geometry shown in Fig. 2a. As we will
show in our dispersion diagrams in Section IV, the SLWG structure can be
potentially designed to operate as a BWO due to the backward waves that are
exhibited where the beamline interacts with an SIP or smooth-TIP. This
behavior is also illustrated by the intersection of the inflection point
(solid brown curve) and the beamline (dashed red line) in the dispersion
diagram of Fig. 3a. Furthermore, since the guided electromagnetic modes of the
SLWG structure are distributed over a larger cross section than a conventional
SWG due to the two lateral waveguides that couple to the middle SWG, the power
handling capability of this structure may be enhanced. The SLWG structure is a
serpentine waveguide SWS which is sandwiched between two straight, parallel
rectangular waveguides with similar broad wall dimensions. A single beam
tunnel is through the the center of the SWG structure. The structure resembles
a ladder due to the rung-like appearance of transverse serpentine sections
running between the parallel straight waveguides. The parallel waveguides are
each weakly coupled periodically to the SWG at the middle of each bend by
small rectangular slots, as shown in the longitudinal cross section of Fig. 1a
and in the dimensional markup of the unit cell, Fig. 2a. This waveguide
structure was inspired by a similar microstrip three-way periodic structure
which also exhibits SIPs [28]. However, our structures are only able to
exhibit SIPs in the fundamental $\mathrm{TE_{10}}$rectangular waveguide mode
by breaking glide symmetry at this time.

The dispersion curves corresponding to the $\mathrm{TE_{10}}$ mode of the
individual parallel rectangular waveguides must be bent and vertically shifted
in order to obtain an SIP or TIP of a desired tilt and frequency in the SLWG
dispersion relation, as will be further described in Sec. III-C.

Introducing small dimensional differences between the top and bottom parallel
waveguide ways (i.e., changing the broad wall dimensions $a_{t}$ and $a_{b}$
as shown in Fig. 2a) or introducing metallic obstacles of different dimensions
in the top and bottom parallel waveguides breaks glide symmetry. Breaking
glide-symmetry provides a simple route to achieve a cold SIP in the
structure’s dispersion relation once periodic coupling is introduced between
individual-waveguide modes. These differences between the top and bottom
waveguides directly control the tilt of the TIP. In general, breaking glide
symmetry is not a necessary condition to have an SIP, as three-way waveguide
structures with unbroken glide-symmetry have been previously shown to exhibit
SIPs [28]. However, here it is has been found to be convenient to achieve and
manipulate SIPs in structures with broken glide-symmetry, as the SIP can occur
within the fundamental $\mathrm{TE_{10}}$ mode of the SWG below upper band-
edge associated to the bandgap of the SWG which occurs at $k=2\pi p/d$ .

###  III-B Three Coupled Serpentine Waveguide (TCSWG)

The three-coupled serpentine waveguide TCSWG structure is our second example
of a serpentine-type SWS that is capable of exhibiting SIPs in its dispersion
relation, and has its geometry shown in Fig. 2b. The TCSWG seems to be better
suited for use in a TWT than the SLWG structure, since the example we show
tends to exhibit forward waves in proximity to the synchronization point where
the beamline interacts with an SIP or a smooth-TIP, as we will show in our
dispersion diagrams in Section IV. The TCSWG structure is constructed
similarly to the SLWG structure. However, the top and bottom rectangular
waveguides which sandwich the center SWG are also made to be serpentine in
shape, giving them a longer path length and similar dispersion shape to that
of the center SWG, as shown in the longitudinal cross-section of Fig. 1b and
in the dimensional markup of the unit cell in Fig. 2b. No periodic obstacles
or broad wall dimension variations are required, as the mean path lengths of
each individual waveguide way may be altered to break glide-symmetry and
obtain an SIP. In our TCSWG structure, the top, bottom, and middle SWGs each
have the same pitch and broad wall dimension. The straight section height of
each SWG way ($H_{t}$, $H_{m}$, $H_{b}$) is varied to alter the shape of its
respective dispersion curve, allowing the prementioned SIP conditions to
occur. The tilt of the TIP is controlled by the size of the coupling slots
placed between bends of adjacent SWSs, as well as minor path length
differences (between the serpentine height dimensions $H_{t}$ and $H_{b}$)
introduced to break glide symmetry. We have also found that if the path length
of the middle SWG, mainly controlled by $H_{m}$, is made to be a scalar
multiple of the top or bottom path length that is greater than two, it is
possible to have multiple SIPs or TIPs in synchronism with the beamline at
different frequencies, though we do not show it in this paper.

Instead of inserting a beam tunnel only in the center SWG, it is also possible
to add beam tunnels to the center of the top and bottom SWGs. This makes the
structure operate with two beams propagating parallel to each other, provided
that the two beams are not too close together and an external magnetic field
can be used to confine both beams. This dual-beam structure can potentially
benefit from increased power output at the SIP/TIP due to beamline synchronism
in both the top and bottom SWG sections.

###  III-C Design methodology

A minimum of three ways is required to obtain an SIP in a reciprocal,
lossless, cold structure. This is because the SIP is a synchronous coalescence
of three eigenmodes. In order to design three-way serpentine waveguide SWSs
that are both consistently synchronized to a beamline and exhibit SIPs, we
utilize a design methodology based on the work of [52]. We use a dispersion
approximation for the initial design of both the individual straight
rectangular waveguide and serpentine waveguide ways. The design process begins
by selecting a fixed center operating frequency, spatial harmonic number $p$,
cell pitch $d$, and average electron beam velocity $u_{0}$ determined from the
cathode-anode voltage of an electron gun. The full-cell pitch $d$ in our work
is chosen to be equal to $\lambda_{g}/4$, where
$\lambda_{g}=2\pi/\left(k_{0}\sqrt{1-c/(2af_{\mathrm{center}})^{2}}\right)$ is
the guided wavelength at the center operating frequency within the SWG
containing the beam tunnel, where $k_{0}$ is the free space wavenumber, $c$ is
the velocity of light in free space, $a$ is the broad-wall dimension of the
individual waveguide cross-section as shown in Fig. 2, and
$f_{\mathrm{center}}$ is the center operating frequency .

Starting with the average beam velocity, $u_{0}$, the beamline’s linear
relation (neglecting space charge effects) between the frequency and average
electronic phase constant , $\beta_{0}$, is

$\beta_{0}=\frac{2\pi f}{u_{0}}.$ (4)

Then, the dispersion for the $p^{th}$ spatial harmonics of the individual
serpentine and/or straight waveguide sections is calculated from the relation
found in [52]

$f=f_{c}\sqrt{1+\left(\frac{a}{L}\right)^{2}\left(\frac{kd}{\pi}-2p\right)^{2}},$
(5)

where, $f_{c}=c/(2a)$ is the cutoff frequency of the rectangular waveguide
cross-section and $L=2H+\pi d/2$ is the mean path length of the individual
uncoupled waveguide section within the unit cell, as can be observed in the
serpentine waveguide sections of Fig. 2. To model the dispersion of a straight
rectangular waveguide sections of Fig. 2a, the path length simply becomes
equal to the pitch of the unit cell, $L=d$. We describe our structure using a
full unit cell notation (of period d) rather than the half unit cell notation
(of period d/2) commonly used in literature. The half-cell notation is often
used because the beam “sees” two beam tunnel intersections per geometric
period of the full unit cell, which only differ by a sense-inversion of
$E_{z}$ fields at each consecutive beam tunnel intersection. The primary
difference is that the path lengths of each individual waveguide of our full
unit cell are twice as long as they would be in the half-cell notation. Much
of the fundamental spatial harmonic of the full-cell dispersion diagram lies
above the light-line of $k_{0}=\pm\omega\sqrt{\mu_{0}\epsilon_{0}}$ and cannot
be utilized for amplification without the use of a relativistic beam velocity
[2]. Because the full-cell notation is being used in this paper rather than
the half-cell notation, the additional $\pi$ phase shift considered in some
other papers such as [52, 53, 54, 55] is no longer needed, so the term $2p+1$
in [52] has been replaced with $2p$ in (5). As long as the coupling between
waveguide sections in each unit cell is weak, this dispersion relation will
serve as a reasonably accurate approximation of actual dispersion below the
first $k=2\pi p/d$ bandgap, which occurs at the intersection of two
neighboring spatial harmonic curves corresponding to the same individual
waveguide way which contains the beam tunnel.

The dimensions $a$ and $H$ of the SWG sections containing beam tunnels, as
shown in Fig. 2, are selected using an optimization algorithm which minimizes
the integrated frequency error between the beamline and SWG dispersion curves
over the wavenumber interval of $kd/\pi\in\left[2p,2(p+1)\right]$. This
wavenumber interval corresponds to the frequency range over which
amplification is desired to occur in our paper, as shown in Fig. 3a and Fig.
3b. Once suitable $a$ and $H$ dimensions are determined for the SWG with a
beam tunnel, the narrow wall $b$ dimension of our serpentine waveguide way was
chosen to be $b=a/6$ to provide adequate spacing between the SWG beam tunnel
intersections. Of course, in most SWG structures, the a and b dimensions are
rarely close to standard waveguide sizes due to the need for synchronism with
a specific beamline, so waveguide transitions are needed at the input and
output ports to allow connections for standard waveguide sizes, in addition to
radio frequency (RF) windows to maintain vacuum within the tube. However, for
simplicity, we do not consider such waveguide transitions or windows in our
study, and we only focus on the beam-wave interaction region. The beam tunnel
diameter $d_{t}$ is selected based on the empirical formula from [52],

$d_{t}=L\alpha\left(1+\left(\frac{L}{2a}\right)^{2}\right)^{-\frac{1}{2}},$
(6)

which minimizes the width of the bandgap caused by the beam tunnel. The ratio
of the beam tunnel radius to the free space wavelength at the $2\pi$
frequency, $\alpha\simeq 0.115$, was used in the design of our structures as
well. The bandgap normally caused by the beam tunnel is significantly widened
due to the additional periodic reactive loading introduced by coupling slots.
Increasing the size of the coupling slots introduces stronger coupling between
the waveguide sections, but enlarges the bandgap. Therefore, simply having a
beam tunnel which is sufficiently in cutoff appears to be adequate for these
kinds of structures. A square-shaped beam tunnel with side length $d_{t}$ may
also be used in place of a conventional cylindrical beam tunnel, as shown in
Fig. 2, to make the structures more compatible with multi-step LIGA
(lithographie, galvanoformung, abformung; German for lithography,
electroplating, and molding) processes [56]. Traveling wave tube amplifiers
with square beam tunnels may be fabricated using two-step LIGA processes like
in [57, 58, 59, 60, 61, 62, 63]. In the two-step LIGA process, the SWSs are
electroformed out of two symmetric halves which are later bonded together.
However, more than two steps will likely be necessary for our structures due
to the coupling slot lengths differing from the beam tunnel width or differing
broad wall dimensions in each waveguide way. Additionally, the use of a square
beam tunnel may slightly degrade the hot operation of TWTs, as mentioned in
[64, 61, 60]. While it is potentially challenging to fabricate such structures
using LIGA fabrication, it should not be significantly more challenging than
it already is for conventional serpentine waveguides fabricated by two-step
LIGA. For example, each additional LIGA step required for the SLWG and TCSWG
structures corresponds to a repetition of procedures 7-13 (lapping/polishing,
photoresist attachment, mask, nth layer alignment, exposure, development, and
electroplating) after procedure 13 shown in [61].

The dispersion condition of the waveguides (when uncoupled to each other)
utilized by our group to consistently obtain SIPs is to have two nearly
parallel uncoupled (individual-waveguide) modes cross over a third
(individual-waveguide) mode which is nearly perpendicular to the other two
modes on the dispersion diagram, as shown in Fig. 3. If a periodic coupling is
introduced between all three of the individual-waveguide modes and the nearly
parallel individual-waveguide modes are in close proximity in the dispersion
diagram, then two phase- and frequency-shifted bandgaps will form at the
intersection points. If the top band-edge of one bandgap tangentially touches
the bottom band-edge of another bandgap, an inflection point is able to form
between these band-edges, as illustrated in the inset of Fig. 3a and Fig. 3b.
Varying the proximity of near-parallel individual-waveguide modes for a given
coupling strength directly controls the tilt of the TIP. Near-parallel
individual-waveguide modes which are close to each other tend to form smooth-
TIPs, whereas near-parallel individual-waveguide modes which are further from
each other will typically form alternating-TIPs. Between smooth-TIP and
alternating-TIP conditions, an SIP condition is expected to exist.

Once the basic dimensions of the SWGs with beam tunnels are established,
coupling slots are positioned between the bends of adjacent waveguide ways to
periodically couple the individual-waveguide modes. The coupling slots have a
vertical thickness $t$ in the y-direction, width $w$ in the z-direction, and
length $l$ in the x-direction, as shown in Fig. 2. The length of the coupling
slot, which is also along same axis as the broad wall dimension of the
waveguide ways, is the dimension that strongly controls the evanescent
coupling of modes between waveguide ways. The width of the coupling slot
controls the wave impedance within the slot, and the slot thickness primarily
controls the extent of evanescent decay and phase delay for waves which are
below the slot cutoff frequency. In this paper, we use a coupling slot length
equal to half the $a$ dimension of the SWG section containing the beam tunnel.
The slot width and thickness are arbitrarily chosen in this paper to
demonstrate that SIPs can be attained in our structures. Larger slot lengths
will strengthen mode coupling, but the dispersion relation of the actual
structure will be strongly dissimilar to the dispersion relation of the
individual uncoupled waveguide modes from before. While large slot lengths and
widths enhance mode coupling, the reflections introduced by the periodic slot
reactance in a finite-length structure may also make the hot structure more
susceptible to regenerative oscillations.

Finally, a small geometric difference is introduced between the waveguide
sections corresponding to near-parallel dispersion curves to control the
frequency and wavenumber spacing between near-parallel dispersion curves. This
directly controls the tilt of the TIP. Large geometric differences typically
result in an alternating-TIP, whereas small geometric differences make the TIP
smooth. For the TCSWG structure, geometric differences may be introduced as a
height difference $\Delta H=\left|H_{t}-H_{b}\right|$ between the top and
bottom serpentine sections ($H_{t}$ and$H_{b}$ in Fig. 2b, respectively). For
the SLWG structure, a broad wall dimension difference $\Delta
a=\left|a_{t}-a_{b}\right|$ may be introduced between the top and bottom
straight waveguide sections ($a_{t}$ and $a_{b}$ in Fig. 2a, respectively).
Alternatively, geometric differences may be introduced in periodic capacitive
obstacles loading the top and bottom straight waveguides of the SLWG structure
to achieve the same effect, reducing the number of steps required for LIGA
fabrication. However, it may not be desirable to use periodic obstacles due to
the large reflections they introduce in the top and bottom waveguide sections
of the SLWG structure. Conversely, the use of unloaded parallel waveguides
also reduces the complexity and reflection coefficients of SWS at the cost of
more steps with LIGA fabrication.

Once all initial structure dimensions are determined, the full unit cell
geometry may be simulated in an eigenmode solver to obtain the Pierce
(interaction) impedance and $\omega-k$ dispersion relation with real-valued
wavenumber, $k$. In this paper, we use the software CST Studio Suite to obtain
the modal dispersion for each periodic structure. Using simulated dispersion
data, the $a$ and $H$ dimensions of SWG ways containing beam tunnels are then
further tuned to recover beamline synchronism. Adjusting the $a$ dimensions of
each serpentine waveguide shown in Fig. 2a ($a_{m}$, for the middle SWG of the
SLWG structure; $a_{t}$ and $a_{b}$ for the top and bottom SWGs of the TCSWG
structure) primarily serves to shift each respective SWG dispersion curve up
or down. Adjusting the $H$ dimension primarily controls the slope of the SWG
dispersion curve, which may have changed due to periodic loading from the
slots. When the modal dispersion curves and TIP are satisfactorily
synchronized to the beamline, the geometric difference between waveguide
sections corresponding to parallel dispersion curves (for example, $\Delta a$
in the SLWG structure or $\Delta H$ in the TCSWG structure) may then be tuned
to adjust the tilt of the TIP to make it close to an SIP.

(a)

(b)

Figure 3: Dispersion for the individual (uncoupled) waveguides in (a) SLWG
structure with a small difference between the broad-wall widths of top and
bottom waveguides $\Delta a=\left|a_{t}-a_{b}\right|$ (blue and green curves
for the straight waveguides, black for the SWG, and dashed red curve for the
beamline) and (b) TCSWG structure with a small difference between serpentine
heights of top and bottom SWGs $\Delta H=\left|H_{t}-H_{b}\right|$ (blue and
black curves for the top and bottom SWGs, magenta for the center SWG, and
dashed thin red curve for the beamline). Each solid curve corresponds to the
dispersion of an individual-waveguide way based on Eqn. 5. The width and
cutoff frequency of each dispersion curve can be set by adjusting the
corresponding geometry for the individual-waveguide way. Insets of figures (a)
and (b) illustrate SIP formation when coupling is introduced (thick brown
curves, dotted brown lines for tilted bandgaps). Introducing coupling between
the waveguides causes bandgaps to form at each dispersion curve intersection.
If two nearly-parallel individual-waveguide dispersion curves cross over a
third individual-waveguide dispersion curve, it is possible to obtain an SIP.
The SIP or TIP occurs where the lower band-edge of one bandgap meets the upper
band-edge of an adjacent bandgap. Tilted inflection point “tilt” may be be
adjusted by varying coupling strength or by adjusting the distance between
parallel individual-waveguide dispersion curves. Dimensions for this example
are provided in Appendix A.

##  IV Results

Following the aforementioned design procedures of the previous section, we
obtain the real part of the modal dispersion relation (for the lossless and
cold structures shown, imaginary parts of the dispersion relation correspond
to evanescent modes, e.g. below the cutoff frequency of the waveguide or in
bandgaps where neighboring spatial harmonics meet on the dispersion diagram)
for both SLWG and two-beam TCSWG structures, as shown in Figs. 4a and 4b,
obtained using the methods shown in Appendix B and verified using the
eigenmode solver in CST Studio Suite. In the insets of these figures, we also
demonstrate how it is possible to vary the tilt of the inflection point for
three different cases simply by altering the difference in straight waveguide
widths ($a_{t}$ and $a_{b}$ for top and bottom rectangular waveguides,
respectively) for the SLWG structure, or the serpentine waveguide heights
($H_{t}$ and$H_{b}$) of the top and bottom waveguides. The dimensions of each
case are provided in Appendix A. From the dispersion relation shown for the
SLWG in Fig. 4a, the point where the beamline intersects an SIP or smooth-TIP
(solid black and dashed blue curves, respectively) is a backward-wave
interaction, making the SLWG design better suited for use in a BWO rather than
a TWT. While an alternating TIP (magenta dotted curve in the inset of Fig. 4a)
might enable forward-wave interactions in the SLWG structure, the upper and
lower band-edges on either side of the inflection point still pose a
significant risk for oscillations. Backward wave oscillators constructed with
the SLWG structure may also benefit from improved power handling capability
compared to a conventional SWG BWO due to the guided electromagnetic mode
being distributed over a larger cross section in the two lateral waveguides.

From the dispersion relation for the TCSWG structure in Fig. 4b, the point
where the beamline intersects the inflection point is a forward wave for the
SIP and smooth-TIP (solid black and dashed blue curves, respectively), making
the TCSWG structure a better choice for TWT designs. While the TCSWG structure
shown is designed for velocity-synchronism with an electron beam at the
inflection point, which is inherently a narrowband phenomenon, this does not
mean that the bandwidth of TWTs built using the TCSWG structure is severely
limited. Because we initially designed these structures with broadband beam-
wave synchronization in mind, there are still broad frequency ranges along
dispersion branches above and below the designed inflection points, where the
beamline is velocity-synchronized to the guided waves. It is still possible to
amplify waves over broad bandwidths as long as the TWT is stable when the beam
is introduced.

Additionally, metal losses and manufacturing errors may perturb the dispersion
relation of an SIP or TIP from its intended design, so such effects need to be
considered in the fabrication and testing of SLWG and TCSWG structures.
Waveguide structures exhibiting EPDs are known to be quite sensitive to losses
and fabrication tolerances[23, 26, 27, 28]. For the example SLWG structure in
this paper, the geometric difference in $\Delta a$ needed to go from an SIP to
a smooth/alternating TIP in the inset of Fig. 4a is $30\ \mathrm{\mu m}.$ For
the example TCSWG structure, a difference in $\Delta H$ of $50\ \mathrm{\mu
m}$ is needed to go from an SIP to a smooth/alternating TIP in the inset of
Fig. 4b. The work of Li, _et. al._ [58] shows that it is possible to fabricate
serpentine waveguide structures with $6\ \mathrm{\mu m}$ depth tolerance (for
the broad-wall dimension $a$ of our structures) and $2\ \mathrm{\mu m}$ width
tolerance between sidewalls using ultraviolet LIGA (UV-LIGA) fabrication
techniques. A fabrication tolerance of $6\ \mathrm{\mu m}$ (20% of the
difference in $\Delta a$ needed to go from an SIP to a smooth/alternating TIP
shown in Fig. 4a for the SLWG structure) is acceptable to obtain a SLWG
structure that exhibits a nearly-stationary inflection point. For the TCSWG
structure, a tolerance of $2\ \mathrm{\mu m}$ for sidewall widths (4% of the
difference in $\Delta H$ needed to go from an SIP to a smooth/alternating TIP
shown in Fig. 4b for the TCSWG structure) is even better. Though, of course,
these tolerances may be relaxed for designs that have an SIP/TIP at lower
frequencies.

Due to the neighboring upper and lower band edges in proximity to the
inflection point, one must also carefully “aim” the beamline, which is
directly controlled by the accelerating voltage of the electron gun, to avoid
striking dispersion branches that have zero group velocity, such as the band
edge, as it can lead to instability [43, 44, 45, 46]. Because there are upper
and lower band edges at frequencies close to the inflection point, the
tuneability of the beam voltage is limited. For instance, in Fig. 4a,
neglecting space charge (i.e. at low beam currents), the average beam velocity
can only be varied by approximately $u_{0}=0.200c\pm 0.001c$ to avoid striking
the neighboring upper or lower band edges on other dispersion branches. This
beam velocity range corresponds to an approximate kinetic equivalent voltage
tuneable range of $V_{0}=10.54\pm 0.11\ \mathrm{kV}$ from the relativistic
relation $V_{0}=c^{2}/\eta\left[1-\left(u_{0}/c\right)^{2}\right]^{-1/2}$in
[65], where $\eta$ is the charge-to-mass ratio of an electron at rest. For the
case of the TCSWG structure we show, the tuneable range of the beam velocity
equivalent kinetic voltage is better due to the neighbouring upper/lower band
edges near the inflection point being separated at higher/lower frequencies,
respectively, as can be observed in Fig. 4b. However, there is still a risk of
oscillations at the lower band edge corresponding to a frequency of approx.
133 GHz for the structure shown, so the tuneable range of beam velocity for
the TCSWG structure is $u_{0}=0.300c\pm 0.002c$, which corresponds to an
approximate beam voltage range of $V_{0}=24.67\pm 0.36\ \mathrm{kV}$.

One must also consider how the electron beam perturbs the inflection point in
the hot dispersion relation for the three-mode synchronization regime, as was
explained in [22]. When an electron beam is introduced to the system, the SIP
or TIP which existed in the cold dispersion relation will be deformed in the
hot dispersion relation. The amount that the electron beam perturbs the
inflection point depends primarily on the dc beam current density, with higher
currents causing larger perturbations to the inflection point in the hot
dispersion relation.

(a)

(b)

Figure 4: Modal dispersion diagrams for (a) SLWG unit cell, and (b) TCSWG unit
cell, with beamline (red dashed). The insets show a smooth-TIP, alternating-
TIP, and SIP in blue (dashed line), magenta (dotted line), and black (solid
line), respectively for each structure. Only wavenumbers which are purely real
are shown. The dispersion diagrams in the figure are obtained using the
methods shown in Appendix B, and the dispersion relations were verified using
the full-wave eigenmode solver of CST Studio Suite. Tilting of the inflection
point is achieved by fine-tuning one or more of the structure dimensions.
Dimensions for the smooth-TIP, alternating-TIP, and SIP are available in
Appendix A.

Similar to other three-way waveguide power dividers and combiners explored in
papers such as [66, 67, 68], our structures also exhibit directional coupler-
like behavior at its SIP/TIP frequencies, as demonstrated with a finite-length
structure of 32 unit cells, shown in Figs. 5a and 5b. The S parameters of the
finite length structure were calculated using the methods explained in
Appendix B. The port numbering scheme for our structures is that the input
ports at the electron-gun end of the structure (on the left) are odd-numbered
from top to bottom, and output ports at the collector end of our structure (on
the right) are even-numbered from top to bottom. While it is highly important
to consider the effect of waveguide transitions and RF windows, our study
focuses primarily on the interaction region of linear beam tubes, so for
brevity we do not consider the effect of input/output coupling structures;
i.e. we only consider the S-parameters at reference planes between the SWS and
where an RF window would be placed in a fabricated device. This directional
coupler-like behavior enables distributed power extraction (DPE) which can be
directed either backward toward the cathode-end of the structure or forward
toward the collector-end of the structure. However, only forward-directive DPE
may be desired for amplification, due to the potential risk of regenerative
oscillations introduced by amplified waves returning to the electron gun-end
of the structure, like in [69]. The introduction of distributed power
extraction for linear beam tubes was necessary to conceive the degenerate (or
exceptional) synchronization in the hot systems studied in [31, 30, 29]. For
the TCSWG structure, increasing the path length of the middle SWG can allow
forward-directive DPE to occur at certain frequencies and for power to be
extracted in the top, middle, and bottom SWG outputs. However, there may still
be SIP/TIP frequencies where backward-directive DPE continues to occur.
Shifting the frequency above or below the SIP/TIP in the vicinity of the
inflection point directly controls whether the top or bottom SWG section
contributes more power to the output of the middle SWG, as can be seen from
the scattering parameters. This dual beam TCSWG structure may also be excited
either from the middle SWG input or top and bottom SWG inputs to achieve
amplification and DPE at the SIP/TIP frequency if coupling is sufficient.
Increasing the size of coupling slots enhances DPE, however this also
exacerbates reflections in the finite-length structure and increases risk of
BWO. Because these structures can still be designed to be well-matched with
small coupling slots, longer finite-length structures and higher beam currents
may potentially be used before unstable BWO occurs in PIC simulations and hot
testing.

(a)

(b)

Figure 5: SWS with 32 unit cells and S parameters for (a) SLWG and (b) TCSWG
in the vicinity of the SIP with port numbering scheme. Port de-embedding is
shown in red. Scattering parameters are calculated using the method explained
in Appendix B.

Finally, we compute the Pierce impedance as discussed in Section II for a
fourth case of the SLWG structure in the vicinity of a nearly-stationary
smooth-TIP, which has dispersion relation similar to the black curve, but the
inflection point is not as not as tilted as the blue curve, shown in Fig. 4a,
with dimensions provided in Appendix A. We demonstrate the benefit of using
nearly-stationary TIPs to enhance the Pierce impedance of serpentine-like
structures, as shown in Fig. 6. We also compare the Pierce impedance of the
SLWG (solid blue line) to the the Pierce impedance of a conventional
individual SWG (red dotted line) (i.e., the serpentine of the SLWG structure,
with removed coupling slots and straight waveguide ways). We find that the
pierce impedance of the full SLWG structure is several times higher than a
conventional simple serpentine waveguide at the frequency corresponding to a
nearly-stationary inflection point. We also note that, while the interaction
impedance of the SLWG appears quite small relative to the simple SWG at
frequencies beyond the inflection point, the interaction impedance is
comparable that of an SWG on other higher/lower frequency branches of the
SLWG’s dispersion diagram, which are not shown.

Figure 6: Pierce (interaction) impedance (solid blue curve in inset) for the
SLWG calculated from the field along the beam tunnel for the eigenmode with a
TIP, associated to the circled region in the dispersion diagram (solid blue
curve in main figure). Nearly-stationary TIP with negative fundamental $k$
(i.e., in the fundamental Brillouin zone defined here as $kd/\pi\in[-1,1]$) is
circled for the $p=2$ spatial harmonic. The Pierce impedance at the TIP is
compared to the Pierce impedance associated with the eigenmode of an
individual serpentine waveguide with coupling slots and straight waveguide
geometries removed (red dashed). Specifically, the Pierce impedance was
evaluated over the phase interval $kd/\pi\in[-0.53,-0.43]$ for the TIP at the
appropriate spatial harmonic $p=2$.The dispersion diagram of the individual
SWG mode (dashed red) is also shown in the main figure. The dispersion curves
in the main figure were found using the eigenmode solver of CST Studio Suite.

Similarly, we compute the Pierce impedance for the a fourth case of the dual-
beam TCSWG structure in the vicinity of the nearly-stationary smooth-TIP,
which has a dispersion similiar to the black curve, but not as tilted as the
blue curve shown in the inset of Fig. 4b, with dimensions provided in Appendix
A. We demonstrate that the nearly-stationary TIP can be used to ehance the
Pierce impedance in both beam tunnels, as shown in Fig. 7. We compare the
Pierce impedance of the TCSWG structure (solid blue line) to the Pierce
impedance of a conventional individual SWG (i.e. with coupling slots and
adjacent waveguide sections removed) for each respective beam tunnel (dashed
red for the top SWG, dotted magenta for the bottom SWG). We find that, just
like with the SLWG structure, the Pierce impedance is several times higher
than a conventional SWG at the frequency corresponding to the nearly-
stationary inflection point. Interestingly, below the frequency of the TIP,
the interaction impedance in the lower beam tunnel is higher than in the upper
beam tunnel, whereas the opposite occurs at frequencies above the TIP. Since
glide symmetry is slightly broken due to the top and bottom SWGs having
different $H_{t}$ and $H_{b}$ dimensions, respectively, the dispersion
branches of the individual SWGs (red dashed line for the top SWG and magenta
dotted line for the bottom SWG) are dissimilar. Due to broken glide symmetry,
the peak values of interaction impedance in the top and bottom tunnels are
also different at the inflection point. If an electron beam is introduced to
the SLWG or TCSWG structures and the beam is velocity synchronized to the
SIP/TIP, we say that the electron beam is synchronized to three degenerate
modes, i.e., we have three-mode synchronization, as was described in [22].
Under the three-mode synchronization regime, the Pierce gain parameter $C$
will also become larger than that of a conventional SWG TWT due to the
enhanced interaction impedance.

Figure 7: Pierce (interaction) impedance (solid blue curves) for the top beam
tunnel (inset with black dashed border) and bottom beam tunnel (inset with
black dotted border) of the TCSWG structure for the eigenmode with a TIP,
associated to the circled region in the dispersion diagram showing a nearly-
stationary TIP with negative fundamental $k$ in the $p=2$ spatial harmonic
(solid blue curve in main figure). The Pierce impedance at the TIP is compared
to the Pierce impedance associated with the eigenmodes of the individual
serpentine waveguides at the top of the structure (dashed red) and at the
bottom of the structure (dotted magenta) when the coupling slots and other
waveguide sections are removed. Specifically, the Pierce impedance in both
tunnels of the TCSWG structure were calculated over the phase interval
$kd/\pi\in[-0.56,-0.35]$ for the TIP at the appropriate spatial harmonic
$p=2$. The dispersion branches of the individual SWG modes associated to the
top SWG (dashed red) and bottom SWG (dotted magenta) are also shown in the
main figure. The dispersion curves of the main figure were found using the
eigenmode solver of CST Studio Suite.

##  V Conclusion

We have showcased two novel dispersion-engineered three-way SWSs for use in
linear electron beam devices: the SLWG and TCSWG geometries. Such geometries
are capable of exhibiting SIPs or TIPs in their dispersion relations, and
larger Pierce (interaction) impedance than that of a conventional serpentine
waveguide at the frequency corresponding to the inflection point. Using our
design methodology, we were able to demonstrate simple conditions which enable
one or more SIP/TIP to occur in a three-way waveguide periodic structure once
weak periodic coupling is introduced between individual waveguides. We have
shown the first known example of a millimeter-wave SWS for linear beam tubes
which exhibits stationary or nearly-stationary inflection points in its
dispersion relation. A previous example of a waveguide which exhibits an SIP
at radio frequencies was demonstrated using microstrip technology in [28], and
was the inspiration for this paper.

What is of interest in both of our introduced structures is that the group
velocity in the vicinity of the SIP/TIP may be easily controlled by slightly
breaking glide-symmetry in our geometries. With weak coupling between
waveguides, the dispersion relation of the introduced structures will not be
significantly different from the dispersion relations of the individual
(uncoupled) waveguides. Due to the “three-mode synchronization” regime, the
Pierce impedance, and consequently the Pierce gain parameter, can be
drastically enhanced over narrow bandwidths near the SIP/TIP, when compared to
a conventional serpentine waveguide commonly used for millimeter-wave TWTs and
BWOs. We believe that the introduction of the three-mode synchronization
regime in such structures may enable the design of more efficient, compact
linear beam tubes, as it was speculated in [22]. There is a lot of room for
improvement if one wishes to focus on improving the interaction impedance or
bandwidth further using SIPs. However, we believe that the design methodology
shown in this paper is still useful for designing realistic millimeter-wave
SWSs with inflection points in their dispersion relation for use in linear
beam tubes.Additionally, we have shown how to obtain TIPs with either backward
or forward mode interactions by simply by varying how much glide-symmetry is
broken in our structures, potentially enabling the design of either BWOs or
TWTs from the same initial geometry. We have also showcased the directional
coupler-like properties of both the TCSWG and SLWG structures at frequencies
near-to the SIP/TIP. This property may be exploited in the design of
specialized TWTs or BWOs which can be simultaneously excited from multiple
ports or simultaneously drive multiple loads.

##  Appendix A Dimensions used in Figures

TABLE I: SLWG and TCSWG dimensions used in Figures 4a and 4b. Dimensions are in $\mathrm{mm}$ unless specified otherwise |  Dimensions used for SWLG |  Dimensions used for TCSWG |  Notes
---|---|---|---
$a_{t}$, $b_{t}$ |  $1.879-\Delta a/2$, $a_{t}/2$ |  1.726, 0.432 |  Cross-section dimensions for top waveguide
$a_{m}$, $b_{m}$ |  1.879,$a_{m}/6$ |  1.726, 0.432 |  Cross-section dimensions for middle waveguide
$a_{b}$, $b_{b}$ |  $1.879+\Delta a/2$, $a_{b}/2$ |  1.726, 0.432 |  Cross-section dimensions for bottom waveguide
_l_ |  0.940 |  0.863 |  Rectangular slot length along x-dimension
_W_ |  0.100 |  0.259 |  Rectangular slot width along z-dimension
_t_ |  0.100 |  0.050 |  Rectangular slot thickness between waveguide bends
$d_{t}$ |  0.282 |  0.259 |  Width of square beam tunnel
_d_ |  0.989 |  1.363 |  Unit cell pitch
$H_{t}$, $H_{m}$, $H_{b}$ |  $H=$0.784 |  $0.472+\Delta H/2$, 1.312, $0.472-\Delta H/2$ |  Height of straight sections for top, middle, and bottom SWGs. SLWG only has one _H_ dimension
$u_{0}$/c |  0.200 |  0.300 |  Average electron beam velocity normalized to the speed of light

For the dispersion of the individual-waveguide ways of the SLWG structure in
Fig. 3a (i.e, without coupling), the dimensions are identical to those in
Table I with $\Delta a=0.1$ mm. , with the exception of $H=0.843$ mm, which
differs from the final $H$ dimension used for shown in Fig. 4a. Note that,
unlike the TCSWG structure, the SLWG structure only has one $H$ dimension.

For the dispersion of the individual waveguide ways of the TCSWG structure in
Fig. 3b, the dimensions are identical to those in Table I with the exception
of the normalized beam velocity being $u_{0}/c=0.20$ and the top, middle, and
bottom SWG heights being $H_{t}=0.518$ mm, $H_{m}=1.161$ mm, and $H_{b}=0.418$
mm, respectively.

Table I reports the dimensions for Fig. 4a and Fig. 4b as follows. In the
inset of Fig. 4a, we have the dimensions used in the SLWG column of Table I ,
with $\Delta a=0.21$ mm for the SIP (black solid lines), $\Delta a=0.24$ mm
for the alternating-TIP (group velocity alternates in sign at frequencies
slightly above or below TIP frequency) (magenta dotted line), and $\Delta
a=0.18$ mm for the smooth-TIP (group velocity does not change sign at
frequencies slightly above or below TIP frequency) (blue dashed line).

In the inset of Fig. 4b, we have the dimensions used in the TCSWG column of
Table I, with $\Delta H=0.15$ mm for the SIP (black solid line), $\Delta
H=0.10$ mm for the smooth-TIP (blue dashed line), and $\Delta H=0.20$ mm for
the alternating-TIP (magenta dotted line).

For the Pierce impedance plot of Fig. 6, we use the dimensions of the SWLG
column of Table I, with $\Delta a=0.20$ mm, which provides an smooth-TIP that
is almost stationary (i.e., it is very close to an SIP). For the Pierce
impedance plot of Fig. 7, we use the dimensions of the TCSWG column of Table
I, with $\Delta H=0.125$ mm, which provides a smooth-TIP that is almost
stationary. We simulated the SLWG and TCSWG structures in the eigenmode solver
of CST Studio Suite 2019 with a step size of $0.5^{\circ}$ in the boundary
phase and a tetrahedral mesh setting of 50 cells per max box edge. We found
that the peak value of the interaction impedance at an SIP or nearly-
stationary TIP for both structures depends greatly on mesh density and step
size of boundary phase.

##  Appendix B T Parameters

Using frequency-domain simulations, it is possible to rapidly and accurately
compute the complex dispersion and approximate finite-length S-parameters of a
periodic structure without an eigenmode solver using the frequency-dependent
scattering parameters (S parameters) of a single unit cell having separable
pairs of ports for each way of the multi-way waveguide structure. The benefit
of computing dispersion in this manner is that it is possible to obtain both
real and imaginary solutions for the Bloch wavenumber, $k$, whereas it is only
possible to obtain the real part of the Bloch wavenumber using the eigenmode
solver of CST Studio Suite. For lossless and cold structures, the imaginary
part of the Bloch wavenumber corresponds to evanescent modes, e.g. below the
cutoff frequency of the waveguide or in bandgaps that form where neighboring
spatial harmonics meet. It is also notably faster to obtain the unit cell S
parameters using the frequency domain solver than it is to directly obtain the
modal dispersion from a phase sweep of the periodic boundary in the eigenmode
solver. Using this method, it was possible for us to tune the geometry of our
structures to obtain beamline synchronism and desired inflection point tilt in
a reasonable time frame. The real dispersion obtained by this method was found
to be in excellent agreement with CST Studio Suite’s eigenmode solutions.

The method we use to obtain the complex dispersion and approximate finite-
length S parameters of our periodic structures involves converting between S
parameters and scattering transmission matrices (T parameters) in intermediate
steps. T parameters may be directly obtained through algebraic manipulation of
each S parameter, as both are defined in terms of the same $a$\- and $b$\-
waves.

$\left[\begin{array}[]{c}b_{1}\\\ b_{2}\\\ b_{3}\\\ b_{4}\\\ b_{5}\\\
b_{6}\end{array}\right]=\underline{\mathrm{\mathbf{S}}}\left[\begin{array}[]{c}a_{1}\\\
a_{2}\\\ a_{3}\\\ a_{4}\\\ a_{5}\\\
a_{6}\end{array}\right]\iff\left[\begin{array}[]{c}b_{2}\\\ a_{2}\\\ b_{4}\\\
a_{4}\\\ b_{6}\\\
a_{6}\end{array}\right]=\underline{\mathrm{\mathbf{T}}}\left[\begin{array}[]{c}a_{1}\\\
b_{1}\\\ a_{3}\\\ b_{3}\\\ a_{5}\\\ b_{5}\end{array}\right]$ (7)

By converting our frequency-dependent S parameters to T parameters at each
frequency for the unit cell, it is possible to solve the following Floquet-
Bloch eigenvalue problem at each frequency to obtain the complex dispersion
diagram for the modes of the periodic structure [70, 71, 42],

$\underline{\mathrm{\mathbf{T}}}(\mathit{z+d,z})\boldsymbol{\Psi}(\mathit{z})=e^{-jkd}\boldsymbol{\Psi}(\mathit{z}),$

where, $\boldsymbol{\Psi}$ is the complex state vector for the unit cell
composed of $a$ and $b$ waves at each port, as shown in Eqn. 7 and Fig. 5a and
Fig. 5b, and $d$ is the unit cell pitch. In other words, the wavenumbers of
the fundamental spatial harmonic, may be evaluated directly from the
eigenvalues $\lambda=\exp(-jkd)$ of the T matrix through the relation

$k=\frac{-\ln\left(\lambda\right)}{jd}.$

The modal dispersion diagrams in Fig. 4a and Fig. 4b (only showing the
branches with purely real $k$) are calculated from the eigenvalue problem
shown above and are verified using the eigenmode solver of CST Studio Suite.
It is also possible to estimate the S parameters of our periodic structure
with finite length by cascading T matrices and converting the resultant
parameters back into S parameters using the same algebraic manipulation as
before [72]. This is how the finite-length S parameters shown in Fig. 5a and
Fig. 5b were computed. This method may be readily generalized for 2N-port
periodic structures. It is also possible to solve the eigenvalue for spatial
harmonics other than the fundamental, as the solutions $k_{n}$ are periodic as
$k_{n}=k+2\pi n/d$, with $n=0,\pm 1,\pm 2,...$.

Due to the presence of periodic coupling slots in both of our unit cell
designs, our single-cell frequency domain model for our structures must be
slightly modified in order to avoid placing ports at coupling slots or in
waveguide bends. Our unit cells were modified for the frequency domain solver
by horizontally shifting the reference planes of all ports by $d/4$ along each
waveguide path and adding de-embedding to the ports to account for small
reflections caused by transitions between straight waveguide sections and
waveguide bends and slots, as shown in Fig. 8a and Fig. 8b. This modification
does not appear to significantly affect the dispersion relation of the
periodic structure under study.

(a)

(b)

Figure 8: (a) SLWG and (b) dual-beam TCSWG vacuum unit cells used for
calculating the T parameters from the single-cell S parameters. Port reference
planes are shifted $d/4$ along each waveguide path to prevent ports from being
placed too close to the coupling slots. Port de-embedding is shown in red.
Knowledge of the T matrix allows for fast and accurate calculation of the
scattering parameters of multi-way waveguides with several unit cells.

## Acknowledgments

This material is based upon work supported by the GAANN Fellowship, by the Air
Force Office of Scientific Research under Award FA9550-18-1-0355 and under the
MURI Award FA9550- 20-1-0409 administered through the University of New
Mexico. We thank Dassault Systèmes for providing CST Studio Suite, which has
been instrumental in this work. The authors would also like to thank our
colleague, Mr. Miguel Saavedra, for his assistance in fixing our interaction
impedance calculation.

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