752
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003
Examining by the Rayleigh–Fourier Method the
Cylindrical Waveguide With Axially Rippled Wall
Joaquim José Barroso, Joaquim Paulino Leite Neto, and Konstantin G. Kostov
Abstract—Axially corrugated cylindrical waveguides with wall
radius described by 0 (1 + cos 2
), where 0 is the average radius of the periodically rippled wall with period and amplitude , have been largely used as slow-wave structures in highpower microwave generators operating in axisymmetric transverse
magnetic (TM) modes. On the basis of a wave formulation whereby
the TM eigenmodes are represented by a Fourier–Bessel expansion of space harmonics, this paper investigates the electrodynamic
properties of such structures by deriving a dispersion equation
through which the relationship between eigenfrequencies and corrugation geometry is explored. Accordingly, it is found that for
1 a stopband always exists at any value of ; the con0
dition
0 = 1 gives the widest first stopband with the band
narrowing as the ratio
0 increases. For
0 = 0 5 the stopband sharply reduces and becomes vanishingly small when
0.10. Illustrative example of such properties is given on considering a corrugated structure with
0 = 1 0 = 2 2 cm,
and = 0 1, which yields a stopband of 1.5-GHz width with the
central frequency at 8.4 GHz; it is shown that in a ten-period corrugated guide, the attenuation coefficient reaches 165 dB/m, which
makes such structures useful as an RF filter or a Bragg reflector.
It is also discussed that by varying
0 and we can find a variety of mode patterns that arise from the combination of surface
and volume modes; this fact can be used for obtaining a particular
electromagnetic field configuration to favor energy extraction from
a resonant cavity.
Index Terms—Periodic slow-wave systems, periodic waveguide,
space harmonics.
I. INTRODUCTION
R
ELEVANT to frequency-selective mode reflectors as
well as to high-power microwave generators employing
slow-wave structures, this paper investigates the electrodynamical properties of the sinusoidally rippled wall cylindrical
waveguide. Instead of having attenuation only for a low
frequency below the cutoff value as with an ordinary guide,
the periodically corrugated guide features passband–stopband
characteristics with alternating bands of propagation and
attenuation, which render this structure useful as an electric
filter. And in connection with the periodically varying nature
of the boundary wall, the electromagnetic fields supported by
the corrugated guide result from the superposition of Fourier
Manuscript received November 18, 2002; revised April 9, 2003. This work
was supported by the Research Assisting Foundation of the State of São Paulo
(FAPESP), Brazil.
J. J. Barroso and J. P. L. Neto are with the Associated Plasma Laboratory,
National Institute for Space Research (INPE), 12201-970 São José dos Campos,
Brazil (e-mail: barroso@plasma.inpe.br).
K. G. Kostov is with the Department of General Physics, Sofia University,
Sofia 1164, Bulgaria.
Digital Object Identifier 10.1109/TPS.2003.815482
components, all having the same frequency and group velocity
but with individual amplitudes and phase velocities, with some
waves (or space harmonics) traveling at velocities less than the
speed of light. A conspicuous feature of periodically corrugated
guides, the retardation effect is used in growing-wave tubes,
such as the traveling-wave tube (TWT) and the backward-wave
oscillator (BWO), which require that the electron beam velocity
should be equal (Cherenkov synchronism) or nearly equal to
the phase velocity of the retarded wave. A practical example
of this is found in high-power BWO experiments [1]–[4] that
employ a sinusoidally rippled wall guide as the slow-wave
structure wherein the selected eigenmode consists of slow
space harmonics.
By considering that the electron beam—under the action of a
sufficiently strong external magnetic field so as to propagate in
the axial direction—interacts only with azimuthally symmetric
transverse magnetic (TM) waves, many works [5]–[13] have
discussed the electromagnetic properties of the sinusoidally rippled guide, but giving special emphasis on particular geometries
appropriate to their experiments. Although extensive but overlooking the filtering aspect of such slow-wave structures, these
studies fall short of exploring quantitatively the general question as how the TM -mode critical frequencies rely on the geometry of the corrugated wall. A direct procedure to determine
these frequencies is important in the design of slow-wave structures and also in filtering considerations, as the critical frequencies—located at the edges of the dispersion curve of the periodic
guide—give the breadth and the width of the pass- and stopbands, respectively. In view of this lack of generalization, the
present work presents a comprehensive picture of the electrodynamical properties of the cylindrical waveguide with rippled
by examining how
axial profile
the guide propagation characteristics are altered upon varying
the geometric parameters of the corrugated wall, namely, the
about the
corrugation period and the ripple amplitude
. Through a systematic variation of the corrumean radius
gation variables, results of the frequency parametrization are
displayed in a set of plots, which, in addition to being useful
tools in the design of such slow-wave structures with required
characteristics, make apparent the role played by the corrugation geometry in establishing the propagation properties of the
structure.
On the basis of the Rayleigh hypothesis, this is achieved by
numerically solving the dispersion relation of the sinusoidally
periodic guide, for which the fields are expanded in a space
harmonic series as formulated in Section II that also illustrates
through specific examples the two distinguishable classes of
harmonic fields, i.e., surface and volume harmonics. General
0093-3813/03$17.00 © 2003 IEEE
BARROSO et al.: EXAMINING BY THE RAYLEIGH–FOURIER METHOD THE CYLINDRICAL WAVEGUIDE
properties of the sinusoidally corrugated guide are discussed in
Section III, and to get further insight into the electrodynamic
problem, an analytic treatment follows in Section IV by expanding the dispersion relation up to second-order terms in the
small parameter ; then it is shown that both analytical and nu0.1.
merical methods provide identical solutions as long as
After discussing in Section V the validity of the Rayleigh hypothesis, the accuracy of the results encompassed in this study is
assessed in Section VI through electromagnetic computer simulation of a ten-section corrugated guide acting as a stopband
filter, giving central frequency and bandwidth that closely agree
with those obtained from the Rayleigh–Fourier (RF) dispersion
relation. Summary of the paper and final remarks are contained
in Section VII.
II. DISPERSION RELATION OF THE CYLINDRICAL WAVEGUIDE
WITH SINUSOIDALLY RIPPLED WALL
The corrugated structure that we consider consists of a
cylindrical waveguide with perfectly conducting wall of radius
sinusoidally rippled about the mean radius
, as that
commonly employed in high-power BWO experiments [8], [9],
i.e.,
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electric field tangential to the corrugated surface must vanish,
i.e.,
(6)
Substituting (2)–(3) in (6) and expanding the resulting expres, we find an infinite system
sion in a Fourier integral over
of algebraic equations for the amplitude coefficients
(7)
where
(8)
The existence of a nontrivial solution to the amplitudes of the
space harmonics requires that the determinant of the homogeneous set of (7) be zero. Thus, setting
(9)
(1)
with and defining the amplitude and period of the corrugation, respectively. Due to the spatial periodicity of the structure along the axial coordinate , azimuthally symmetric TM
can
electric fields in the cylindrical corrugated system
be expanded in spatially harmonic series according to Floquet’s
theorem as
(2)
(3)
where
(4)
denotes the longitudinal wavenumber of the th space harand to
monic, which is related to the angular frequency
through
transverse wavenumber
(5)
Assuming as the reference propagation constant, the elecin moving from to
tric field phase shifts by
(at fixed and ), with the determination of , and, hence, ,
as function of the angular frequency being the central problem
of a slow-wave structure.
The dispersion equation that determines the dependence of
on the angular frequency follows
the reference wavenumber
from the boundary condition requiring that the component of the
yields the eigenvalue equation for at given corrugated param, and . If we normalize the dispersion relation and
eters
as a function of
then the periodic strucsolve for
.
ture is described by two dimensionless parameters, and
Although (9) involves an infinite matrix, in practice we truncate the system to an adequate finite rank to obtain an approximation of the exact eigenvalue equation. We have verified that a
9 9 matrix gives (for the same structure) eigenfrequencies differing by 0.1% from those calculated with a higher order matrix.
Therefore, in the ensuing calculations the rank of the matrix in
) is truncated at 9
.
(9) (with
Once the dispersion relation (9) is solved, the ratios of the
are determined from (7). With a known set
coefficients
the field components
and
are calculated from
of
(2) and (3). As an illustrative example of this approach, let us
consider a corrugated waveguide characterized by corrugation
and
. For this strucparameters
(zero phase shift per period), there
ture operating at
corresponds the discrete set of normalized frequencies
, with Fig. 1 displaying
the mode pattern associated with the third eigenvalue. The electric field lines shown in Fig. 1 arise from the combination of
harmonics. To examine further the
nine
harmonic composition in Fig. 1, we combine (4) and (5) to obtain the following relation:
(10)
which quantifies the number of fast space harmonics that propin a periodic
agate at the normalized frequency
structure characterized by the corrugation parameters and
. Using (10) for the third eigenfrequency
we find
if
. This indicates
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003
Fig. 1. Electric field line pattern corresponding to normalized eigenfrequency
!~ = !R =c = 6:9792 at = 0 in a sinusoidally corrugated guide with
ripple amplitude = 0:20 and period-to-average radius ratio L=R = 1.
Fig. 3. Electric field lines at !
~ = 6:2711 for a periodic phase shift ~ = 0:5 and = 0:05.
in a guide with L
L=
Fig. 4. Radial distribution of the electric field axial component at z = 0 in the
corrugated guide of Fig. 3.
Fig. 2. Spatial structure of the norm of the electric field parallel to the tangent
of the rippled boundary in Fig. 1.
that the wave composition in Fig. 1 contains three volumetric
harmonics, including propagating forward and backward
while
components as identified by the indices
the remaining harmonics are all evanescent. For such volume
are real with
modes the transversal propagation constants
the corresponding radial eigenfunctions given by oscillatory
. In fact, we see in Fig. 1 that, due
Bessel functions
to the presence of volumetric harmonics, the resultant axial
electric field vanishes on the cylindrical surface at
and reverses sign twice over a radial excursion from the guide
. This oscillatory feature
axis to the corrugation’s crest at
is also illustrated by Fig. 2, which shows the two-dimensional
of the component of
distribution of the norm
parallel to the tangent drawn through the point
on
the rippled wall, i.e.,
(11)
vanishes on the boundary, thus
In fact, it is apparent that
testifying the effectiveness of the present method in determining
the eigenfrequency and field amplitudes of a set of TM harmonics as required by a corrugated guide with a rippled amplitude as high as 0.20.
Next, let us consider a periodic structure with
at
. In this example, the inequality sign in (10) does
not hold for any integer, and, therefore, no oscillatory components enter into the composition of the required set of harmonics. All the components are slow harmonics, on exception
and
, for both of
of the low-order harmonics with
which the transverse propagation constant becomes zero, since
For these
the equality sign in (10) holds true for
special harmonics, this implies that the associated radial distribution of the electric field is a constant. To illustrate this feagiving
ture, a structure was synthesized with
at
. First, as shown in Fig. 3 the electric
field lines corresponding to families of harmonics
and
with periodic phase shift, respectively, of and
3 are clearly apparent. Nearly uniform over a large radial ex, the radial variation of the resultant electent at the plane
tric field plotted in Fig. 4 demonstrates that the low order com, and, therefore, with
, predomiponents
nate in the bulk of the guide over the higher order
surface harmonics, whose radial distribution is given by the
.
modified Bessel function
Therefore, when we compare with the uniform circular guide,
the periodically corrugated structure may be viewed as a generalized guide that offers an interesting variety of field patterns
arising form the superposition of volume and surface harmonics,
BARROSO et al.: EXAMINING BY THE RAYLEIGH–FOURIER METHOD THE CYLINDRICAL WAVEGUIDE
755
with each component having the appropriate amplitude coefficient and axial wavenumber as required by the boundary condition on the rippled wall. This feature can be explored to give an
adequate field pattern that favors energy extraction from a cavity
resonator operating in the desired mode without reducing the
efficiency of conversion of dc beam to ac output power in microwave generators [14].
In concluding this section, we describe in the following the
method used for generating the field lines of the electric field
. First, we look for a scalar point function
such
that
(12)
Since the gradient
is normal to a curve
, then the contour plots of
i.e.,
lines. From (12)
,
give the field
Fig. 5. Calculated normalized frequencies (filled circles) as function of the
~ = 1:5 and
periodic phase shift L in a sinusoidally corrugated guide with L
= 0:025. Dashed lines are shifted TM-mode dispersion curves for the circular
cylindrical waveguide.
(13)
and noting that
at
we arrive
(14)
Then using the field expressions (2) and (3) in true instantaneous
an exact difform and multiplying (14) by to render
ferential, we finally obtain
(15)
derived in
Therefore, isolevel contours of the function
(15) give a map of the field lines, which can be easily generated,
for instance, by using the built-in command ContourPlot of the
Mathematica package [15].
III. GENERAL PROPERTIES OF THE SINUSOIDALLY
CORRUGATED PERIODIC GUIDE
First, we discuss the dispersion characteristics of a corrugated
with a small but nonvanguide with normalized period
, by illustrating in Fig. 5
ishing corrugation amplitude
a set of frequencies calculated at evenly spaced values of the
, in the range
. Displaced
periodic phase per section,
, also are shown in Fig. 5 dispersion curves versus
by 2
for TM modes in a circular cylindrical guide, that is
(16)
denotes the th root of
. In addition to the
where
close match between frequency points and dispersion curves,
, where dashed curves
we notice on the vertical line
intersect, that a frequency splitting manifests itself as a result
of the interaction of traveling backward and forward waves.
Such frequency splitting due to the coupling of two modes
~ = 1:0 and = 0:15,
Fig. 6. Corresponding to a corrugated guide with L
dispersion curves (solid lines) trigonometrically fitted to calculated points
(filled circles) superimposed to TM-mode dispersion diagram for the circular
cylindrical guide.
through their spatial harmonics becomes more apparent on
considering a larger corrugation parameter so that coupling becomes stronger. Referring to a corrugated guide with
and
, this is shown in Fig. 6 which displays frequency
points that far deviate from the corresponding TM-mode
dispersion curves as frequency goes up. The solid lines give
equally spaced frethe least-squares trigonometric fit to
, namely,
,
quency points in
and
where is a
with fundamental period
. Justification to performing this fitting
whole number
relies on the fact that the guide is symmetrical in both
axial directions and, as such, both
correspond to the
same frequency . Since two field harmonics with propagation
are associated with the same ,
constant
according to Floquet’s theorem [cf. (2)–(4)], then it follows
curve (solid lines) is symmetrical around
that each
and
and must, therefore, have zero
both
slope there. Zero slope implies no power flow along the guide.
No power flow means a resonance condition in each cell, such
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Fig. 7. Electric field line pattern with a phase between cells at half-wave
~ = 1 and
(a) short- and (b) open-circuit resonances in a corrugated guide with L
= 0:15.
that every zero or phase shift frequency is also a resonant
frequency of a unit cell. To illustrate this feature, Fig. 7 shows
the electric field patterns for the third and fourth -resonance
and
, labeled
frequencies at
as the lower and the higher splitting frequencies,
by
,
respectively. Normal to the cross sections at
in Fig. 7(a), the electric field lines (of cosine-like character
) reverse direction at each period, and, thus,
correspond to a half-wave short-circuit resonance with a
phase between cells. In complementary fashion, and as disof sine-like
played in Fig. 7(b), the accompanying mode
character resonates within single cells bounded by magnetic
. In addition, we note that the
and
planes at
frequency points alternate themselves along the
line.
Similarly and having the effect of producing stop bands in
various frequency ranges, frequency splitting also occurs at
. This is illustrated by Fig. 8 which shows the electric
.
field patterns for the third and fourth resonances at
Fig. 8(a) refers to resonance in short-circuited cells where the
amplitudes of forward and backward space harmonics are sym; in the complementary case
metrically related, i.e.,
of open-circuit cell resonance [Fig. 8(b)] the wave components
. Both modes are standing rather
are related by
than traveling waves because a progressive wave, under such
symmetry conditions, is completely reflected as it propagates
along the guide. Regarding a mode that travels along the guide
at the
with a nonzero group velocity, we take
fourth dispersion curve in Fig. 6. This example is given by
and in
the electric field patterns plotted in Fig. 9(a) at
Fig. 9(b) at one-quarter period later. In both plots, the pattern
down the waveguide.
repeats itself at each
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003
Fig. 8. Electric field line pattern corresponding to zero phase shift at (a) short~ =1
circuit and (b) open-circuit cell resonances in a corrugated guide with L
and = 0:15.
Fig. 9. Electric field line pattern corresponding to the propagation constant
= (2=3)=L in a sinusoidally corrugated guide with L~ = 1 and = 0:15
at (a) !t = 0 and (b) !t = =2.
IV. ANALYTICAL DISPERSION EQUATION AND HOW
RELATES WITH THE CORRUGATION GEOMETRY
To derive an analytical dispersion relation for the sinusoidally
corrugated waveguide, we begin by expanding around
and retaining terms of order up to
the Bessel function
in the integrand of coefficients
BARROSO et al.: EXAMINING BY THE RAYLEIGH–FOURIER METHOD THE CYLINDRICAL WAVEGUIDE
in (8). Performing the integral over the structure period,
becomes
then the coefficient
(17)
and
is the Kronecker delta. On the
where
must vanish, cf. (9), and by concondition that
, we arrive at the
sidering three space harmonics
following relation:
(18)
We see that in the limit of a vanishingly small ripple amplitude , (18) reduces to the relation
which can be interpreted as coupling of the zeroth order space
with the first backward
and forharmonic
traveling space harmonics. These solutions have
ward
been represented in Fig. 5 by the corresponding dashed lines,
closely matching the dispersion curves for a corrugated guide
.
with a small ripple of
To verify the degree of accuracy provided by the approximate dispersion equation, we compare for some periodic lengths
and ripple amplitudes the eigenfrequencies calculated at
from (18) with those obtained by the numerical
method formulated in Section 3. This comparison is presented
in the parameterized plots of Fig. 10 which shows as function
as a parameter.
of with
Continuous lines refer to the analytic dispersion relation,
and
whereas scattered points (labeled as triangles for
) are associated with the more accurate, numercircles for
ical approach. It is apparent that for the lower -resonance
, and to be justified later, the anaat the shortest
0.10
lytic dispersion fails to give accurate results for
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[Fig. 10(a)], yet we see that in general both results correlate
quite well with agreement improving as
increases. The
sequence of plots in Fig. 10(a)–(c) clearly demonstrates the
frequency splitting of -modes (open circles) that starts at
and leads to a frequency
that increases with
separation
. In this way
constitutes a stopband, in which no real
power can flow at the operating frequencies lying within
for a chosen . We remark that for longer periodic lengths
, Figs. 10(b)–(d),
stays entirely confined between
curves. By complementarity, a passband is given
the
by the frequency range bounded by adjacent
and
curves. Hence, a passband curve should flatten as increases.
case [Fig. 10(a)], the
For the shorter length
curve—thus
lower -frequency curve overlaps with a
as
producing narrow stopbands even for a high
[Fig. 10(b)], the
shown in Fig. 11(a). However, for
lies halfway the two
coalescing frequency
, i.e.,
zero-phase-shift frequencies at
and
. In this case, the two
at each in comparison
splitting curves produce the widest
cases. As illustrated in Fig. 11(b), the stopwith the
corresponding to
with
extends about
band
frequencies.
half the range between the upper and lower
In the following, we explain why the eigenfrequencies
obtained through the analytic dispersion relation match more
closely those calculated from the numerical formulation as the
becomes longer. Rooted in the expansion
periodic length
up to terms , the discrepancy arises from
of
inappropriately large values of the transverse wavenumber
, because not only the smallness of but also
(over which the approximate function
the range of
is used) controls the accuracy of the approximation achieved.
the approximate
begins to
Quantitatively, for
from the exact function
with increasingly
diverge at
, above 3%; for
relative error, defined here by
, the relative error is as high as
the imaginary value
17.5%. For a corrugated guide with a given , a longer period
implies lower frequency, which translates into smaller
. In
associated with
(at
fact, the wavenumber
and
) is
, a value relatively small that renders
the approximation accurate. However, for the shorter period
, at
and
, the corresponding
assumes a large value for which
wavenumber
the expansion fails, as captured by Fig. 10(a).
V. DISCUSSION
In this section we discuss the validity of representing the
fields in a corrugated waveguide by a single expansion applicable everywhere inside the guide so as to identify some quantitative criteria to check the accuracy of the calculated results.
Then we consider the tangential component of the -th space
harmonic, which from (2), (3), and (6) is written in the form
(19)
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003
Fig. 10. Eigenfrequencies for zero (denoted by triangles) and (open circles) phase shift per period as function of the normalized corrugation amplitude with
~ as a parameter: (a) L
~ = 0:5; (b) L
~ = 1:0; (c) L
~ = 1:5; and (d) L
~ = 2:0. Continuous curves are calculated from the analytic dispersion (18).
the periodic L
This expression is an even periodic function on the interval
whose Fourier expansion coefficients are just given by
of the th column of the matrix
, cf. (8)
the elements
(20)
Fig. 11. First two TM-mode dispersion curves for a sinusoidally corrugated
~ = 0:5; = 0:15 and (b) L
~ = 1:0; = 0:20. In
waveguide with (a) L
~ also is shown the light line ! = c .
normalized form !
~ =( L=) = =L
Through quantitative examples we attempt to identify the
conditions that allow the prescribed analytical function
in (19) to be closely approximated by the harmonic expansion
in (20). With
, this is verified by examining two field
with
: the first, for
,
solutions at
, and the
with corresponding eigenfrequency
for a deeper corrugated guide
second solution
. Curves comparing
with its cosine repwith
resentation (20) are shown in Fig. 12(a) and (b) for
and
, respectively. For the shallow corrugation the
is accurately matched by the cosine expansion
function
[Fig. 12(a)], yet a poor reconstruction is achieved in the deeper
corrugation case.
This is explained by looking at the wavenumber spectrum
for
in Fig. 13(a) which disof waveform
plays as many harmonics (five) as the number of nondegenerate
terms due
terms in the cosine expansion, which provides
. For the deeper corrugation
to the evenness of
, however, the corresponding spectrum [Fig. 13(b)] contains more harmonics than that available (in number of five) for
signal. However, an obvious
reconstructing the original
question can arise. Why not increase the rank N of the matrix
so that a larger number of cosine terms be at hand to re? At high
, however, as the index
construct
is made larger, the shape of the function
becomes increasingly more complex (Fig. 14), as illustrated by
BARROSO et al.: EXAMINING BY THE RAYLEIGH–FOURIER METHOD THE CYLINDRICAL WAVEGUIDE
Fig. 12. Functions T (z ) (continuous curves) for L=R = 1 with (a) 0:10, (b) = 0:20 and the cosine expansions (20), broken curves.
Fig. 13.
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Fig. 14. Functions T (z ) (continuous curves) for L=R = 1 with (a) =
= 0:10, (b) = 0:20, and the cosine expansions (20), broken curves.
Wavenumber spectra of waveforms T (z ) in Fig. 12.
the corresponding wavenumber spectra in Fig. 15, with its harcosine terms. Being
monics outnumbering the available
[Fig. 15(a)], the number of harstill close to 5 at
increases to 10 at
monics in the spectrum of
[Fig. 15(b)] and, as we have verified, this number amounts to
. Then we conclude that for low
12 when
a small suffices for the functions
to be accurately reconstructed, consistent with the fact that the analytic dispersion
well fits the numerical data for
relation (18) derived at
as displayed in Fig. 10.
Here, we deal with a problem different from those usually
by a
found in expanding a known continuous function
of a basis function
through
linear combination
Fig. 15.
Wavenumber spectra of waveforms T (z ) in Fig. 14.
, hoping that
approaches
as closely as we please by taking N large enough. However,
and the shape of
itself
in our problem the coefficients
demanding in a catalytic way a
both depend on , with
number of expansion terms larger than the limit N. Despite this
injunction, we shall see that useful solutions can be obtained
even though the series (2) diverges. First, we note in Fig. 14 that
, continuous line) and the reconstructing
the prescribed (
—the loca(dotted line) curves mostly disagree at
tions where the corrugated profile reaches its maximum radius.
This nicely illustrates a peculiarity of the so-called Rayleigh hypothesis [16]–[21]. Emerged from theoretical considerations of
plane-wave scattering by a rough surface, this hypothesis, postulated by Rayleigh [19], assumes that the scattered field may
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Fig. 16.
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003
Electric field pattern at 1:67=1:5.
= 0 in a guide with = 0:27 and L=R =
be analytically continued to the corrugated surface so that the
scattered mode amplitudes are determined from the boundary
conditions. By following this assumption in the context of the
present problem, the field representation (2)–(3), strictly valid
, is extended to the corrugated cylindrical
for
. As pointed out as by Millar [16] and discussed
surface
by Neviere et al. [22], the validity of the Rayleigh hypothesis
is conditioned by the presence and localization of singularities
of a conformal mapping that transforms a sinusoidal curve into
a straight line. A general explicit criterion for the validity of
the Rayleigh theory is still missing, but limits of convergence
set out by Petit and Cadilhac [23] and Hill and Celli [24] as
0.072 and
0.094 respectively for one- and two-dimensional sinusoidal surfaces render the Rayleigh hypothesis
rigorously valid as addressed by van den Berg and Fokkema
[25]. Further, Berman and Perkins [26] have demonstrated that
the Rayleigh method with boundary conditions being projected
in the Fourier space gives accurate results well beyond the domain of validity of the original Rayleigh hypothesis when applied to a Dirichlet sinusoidal surface, with the method breaking
. As the domain of anadown at deeper grooves
lyticity of the plane-wave Floquet’s expansion depends on the
angle of incidence, such criteria for the sinusoidal grating have
obtained assuming normal incidence. In cylindrical geometry
this translates into an influence of the waveguide radius. Accordingly, we show below that the criterion for the validity of
the RF method when applied to the cylindrical rippled guide,
,
in addition to relying on the ripple-to-periodic length ratio
depends on the waveguide radius as well.
To this end, we consider a sinusoidally rippled guide which
has been manufactured and experimentally tested. Intended for
applications in relativistic BWOs, the guide has the normalized
with average radius
cm
ripple amplitude
cm [9], [12]. With the guide operating
and period
, the correin the second passband at
sponding electric field lines and the associated distribution of
the tangential component of the electric field are shown, respectively, in Figs. 16 and 17, where the tangential field diverges noticeably near the crests of the guide’s profile. However, for this same guide operating (in the second passband) at
, the tangential field neatly matches the
sinusoidal boundary as illustrated by Figs. 18 and 19. Characterized by cross sections magnetically shorted at the maximum
radius of the guide, this kind of field configuration (designated
resonance) favors the RF method in that the elechere as the
tric field lines reach the guide’s crests radially (Fig. 18). And in
configuration in which the eleccomparison with the
Fig. 17. Spatial structure of the electric field parallel to the tangent to the guide
boundary in Fig. 16.
Fig. 18.
L=R
Electric field lines at = 1:67=1:5.
=
=L in a guide with = 0:27 and
Fig. 19. Spatial structure of the electric field parallel to the tangent to the guide
boundary in Fig. 18.
tric field lines curl inside the groove (Fig. 16), the RF method
pattern, which relaxes
proves to work better for the
the calculation from numerical divergence by lacking axial electric field component at the guide’s crest, the location where the
Rayleigh hypothesis becomes weakest.
To quantitatively assess the accuracy of the numerical
, with
solutions, we take the ratio
denoting the electric field, tangent to the
boundary, averaged over a corrugation period, and
the maximum electric field on-axis, as a measure of the error
BARROSO et al.: EXAMINING BY THE RAYLEIGH–FOURIER METHOD THE CYLINDRICAL WAVEGUIDE
761
Fig. 21. Simulation electric vector field plot corresponding to the guide in
~ = 6:9792).
Fig. 1 ( = 0; !
Fig. 20. Error ratios along the boundary of the guides in (a) Fig. 18 and
(b) Fig. 16.
incurred. For the
solution, this ratio gives 2.3% as
becomes maximal at the
indicated in Fig. 20(a), where
and essentially vanishes on the central
period ends
. For the
case, although
region
the error ratio is as high as 1.4 [Fig. 20(b)] the calculated
GHz agrees
frequency
remarkably well—to better than 0.2%—with the experimental
value of 14.29 GHz measured by Guo et al. [12] and also
with that obtained (14.28 GHz) from a state-vector numerical
technique [7], [12].
Regarding some particular features of the RF solutions, we
patterns depicted in Figs. 1–2 and
focus on the two
16–17. Although both solutions correspond to corrugated guides
and
, rewith similar ripple ratios of
diverges catastrophispectively, the solution for
cally in Fig. 17, whereas such a behavior does not unfold at all
, cf. Fig. 2. While having only one volumetric
for
, cf. (10) with
) in
component (
its composition of space harmonics, the remaining components,
, grow exall of which are surfaces harmonics
ponentially faster than the decay of the expansion coefficients
, and so the solution diverges. By contrast, the presence of
facilthree volumetric harmonics in the solution at
itates from the numerical point of view the cancellation of the
tangential electric field along the boundary (Fig. 2). To support
but keeping
this conclusion at a deeper corrugation
with
, we have verified that for
the
holds three volumetric harmonics
solution found
entailing a reasonably small error ratio of 4.0%, which makes
this a useful solution.
To ascertain the field quantities provided by the RF method,
,
we perform on the guide of Fig. 1
by using the code KARAT [27], a time-domain electromagwith the RF-calculated eigenfrenetic simulation at
as a benchmark parameter. In the simuquency
lation, a single short-circuited cell (Fig. 21) with average radius
Fig. 22.
Frequency spectrum of the electric field in the guide of Fig. 21.
cm, and, hence,
GHz, is excited by an ac
current-driven electric probe placed on the cavity axis. With the
input frequency ranging from 15.00 GHz, but kept unchanged in
each run, then by a search procedure the trial frequency is varied
until steadily growing fields are observed over a few hundreds
of the oscillation period in the simulation. The excitation frequency has shown to be slightly dependent on the size of the
square grid used, with 1 2- and 1 5-mm grid sizes giving resonant frequencies of 15.050 and 15.105 GHz, respectively. Howmm the excitaever, decreasing further the mesh size to
tion frequency rapidly converges to 15.145 GHz, as seen in the
frequency spectrum of Fig. 22, in excellent agreement with the
predicted value of 15.147 GHz. We note in addition the obvious
correlation between the simulated electric vector field pattern in
Fig. 21 and the RF-method-calculated field lines as displayed in
Fig. 1.
Finally, we compare electric field profiles obtained from electromagnetic simulation and RF calculation along the radial co[Fig. 23(a)] and running across
ordinate at constant
[Fig. 23(b)]. In its radial exthe cavity at fixed
changes
cursion [Fig. 23(a)], the longitudinal field
exactly lying on
sign twice, with the second zero at
the largest circular cylinder that fits the guide, a picture consistent with the contour plot in Fig. 1. In passing out of the inner
into the groove region
volume
762
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003
~ = 1 and
Fig. 24. Simulation configuration for the corrugated guide (with L
= 0:2) into which a TM wave is injected from the left open boundary
and completely absorbed by the resistive disk on the right termination. Also
shown is the power distribution along the system on steady state; the geometry
is cylindrical with the z -axis corresponding to the axis of symmetry.
Fig. 23. (a) Radial and (b) axial electric field distributions in the guide
of Fig. 21.
both solutions keep agreeing, and only at distance
close to the guide wall
does the RF solution (solid
line) begin to diverge. An accurate fitting of RF to KARAT field
profiles is also obtained in Fig. 23(b), with the RF solution for
departing slightly from the
the radial component
KARAT curve (dashed line) past midway its excursion across
the groove.
VI. FILTERING ASPECTS
In this section, we examine the selective properties of the
sinusoidally rippled waveguide through numerical electromagand
netic simulation of a periodic structure with
cm. As given in Fig. 10, the splitting
average radius
corresponding to short(normalized) frequencies at
and open-circuit resonances are, respectively,
and
which give, according to the normalization
relation
, a stopband in the range 7.7600–9.1716
GHz. Thus, we should expect a strong attenuation at the central frequency of 8.4658 GHz. To verify this we perform, by
using the computer code KARAT [27], a time-domain electromagnetic simulation of the structure shown in Fig. 24. Discretized with square cells 1-mm size, the structure comprises
a corrugated section connected on both ends to a 10-cm-long
smooth-walled guide. At the right of the short-circuited termination, a resistive disk is placed to completely absorb eventual
incident waves without reflection back to the corrugated section.
From the left open boundary a TM wave is launched into the
system at drive frequencies within the 6.5–10.5 GHz range to
ascertain the frequency-response characteristics of the guide in
such a stopband. This is done by determining the attenuation
coefficient from the steady-state power distribution
along the guide, where
is the peak power at
(Fig. 24). Results from this numerical
the injection plane
experiment are displayed in Fig. 25 showing the attenuation as a
function of frequency for a ten-section periodic guide, together
with the attenuation curve calculated from the dispersion relation (9) at complex wavenumber and real frequency. Thus, we
Fig. 25. Frequency-dependent attenuation as determined from simulation
(squares) on a ten-section guide and on an inifinite-length guide from the RF
method (circles).
Fig. 26. Dependence of attenuation on the number of corrugated sections at
8.4 GHz.
see that the maximal attenuation of 165.0 dB/m on the simulation curve occurs at the frequency of 8.4085 GHz, in agreement with the stopband center frequency of 8.4658 GHz as anticipated earlier for the infinitely long corrugated guide. This is
supported further by Fig. 26 giving the attenuation as a function
of the number of periodic sections, where we note that the attenuation curve tends to saturate when the number of sections
increases. This picture is consistent with Fig. 25 as the ten-section curve lies little below the infinite-length guide curve, which
in addition falls from its maximum at a steeper gradient than the
simulation curve does.
BARROSO et al.: EXAMINING BY THE RAYLEIGH–FOURIER METHOD THE CYLINDRICAL WAVEGUIDE
Acting like mirrors in the usual laser cavity, the arrangement
depicted in Fig. 24 constitutes a Bragg reflector, which has been
recognized as an ideal element for free-electron lasers (FEL)
[28] and cyclotron autoresonance masers (CARM) [29]. By providing frequency selective feedback of overmoded high-power
microwave oscillations, the corrugation scatters a forward wave
coherently into a backward wave as long as the corrugation pe.
riod satisfies the Bragg condition, i.e.,
VII. CONCLUSION
This paper has investigated the electromagnetic properties
of the periodic waveguide with sinusoidally varying wall
radius operating in axisymmetric TM modes. As exclusively
considered in BWO experiments, just these modes are able to
perturb the axial velocity and electron density on rectilinear
beams confined by an external magnetic field in slow-wave
systems. Suited for a corrugation geometry of continuously
rippled wall—which is more resistant to RF electric breakdown
when compared with corrugation having rectangular slots
and ridges of sharp edges—the analysis has developed on the
basis of the Rayleigh hypothesis whereby the field solution
is represented by a single expansion of TM eigenmodes with
the boundary conditions projected in the Fourier space. This
so-called RF method has been verified by some authors [18],
[20], [26] as the one (among the least-squares and the integral
methods) giving by far the best overall description of wave
scattering from sinusoidally corrugated surfaces. Demonstrated
here for the cylindrical guide, the RF method has shown
excellent performance as the error incurred in calculating
eigenfrequencies and electric field distributions is small enough
to be disregarded in most practical applications as long as
, noting that the larger the number of volumetric
harmonics, the better the method works. The influence of the
and
corrugation parameters (normalized periodic length
ripple amplitude ) has been carefully investigated in the light
of their causal relation to the dispersion characteristics, with a
parameterized study indicating that the width of the stopband
, where
denotes the normalized
scales as
cutoff frequency of the corresponding smooth-walled circular
waveguide. Furthermore, the favorable comparison of theoretical predictions with computer simulated data for a ten-section
corrugated guide well indicates the suitability of the numerical
tools presented here for the design and analysis of rippled wall
waveguides.
Complementing the numerical results, the analytic dispersion equation so far derived has given frequencies almost coincident with those calculated from the rank-9 matrix formulafor periodic lengths ranging from 0.5 to
tion up to
2 in steps of 0.5. The situation in which the analytical relation
utterly failed to reproduce the numeric-matrix results refers to
with
opguides of short periodic lengths
erating on the lower branch of the -mode splitting frequency
curves, for which decreases with increasing . So long as
over this branch, all the space harmonics are evanes. Then, the transverse wavenumcent waves since
bers all assume imaginary values with the fields growing exponentially in the radial direction. While considering three space
763
harmonics
, this has introduced numerical difficulties as there are no zeros available to meet the required
boundary conditions. However, we remark that devices using
high-current relativistic electron beams demand relatively weak
% as in
wall modulation on the rippled structure (typically
overmoded slow-wave devices), and in this regard the analytic
relation proves to be useful and efficient with respect to both
accuracy and computation time.
Finally, we mention that for triangular and trapezoidal profiles approximated by analytic functions, the RF method still
provides reliable results [20], [25]. Similar to the sinusoidallyrippled waveguide study undertaken here, an investigation on a
variety of corrugated profiles (semicircular, rectangular, trapezoidal, and a combination of semicircles and rectangles [30]) is
currently being considered to examine how the passband characteristics and coupling impedance relate to the shape of the
corrugation.
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003
Joaquim José Barroso received the B.Sc. degree
in electrical engineering and the M.Sc. degree in
plasma physics, both from the Technological Institute of Aeronautics (ITA), São José dos Campos, SP,
Brazil. in 1976 and 1980, respectively. In 1988, he
received the Doctor degree in plasma physics from
the National Institute for Space Research (INPE),
São José dos Campos.
He has remained at INPE since 1982, and has
been involved in the design and construction of
high-power microwave tubes. From 1989 to 1990,
he was a Visiting Scientist at the Plasma Fusion Center, Massachusetts Institute
of Technology, Cambridge, MA. His current interests include microwave
electronics and plasma technology.
Joaquim Paulino Leite Neto received the B.S. degree in physics and the M.S.
degree in plasma physics from the State University of Campinas, Campinas,
Brazil, in 1978 and 1984, respectively.
He has been with the National Institute for Space Research (INPE), São José
dos Campos, Brazil, since 1982, where he has worked on numerical simulation of electron cyclotron resonance heating, Currently, his research activities
concentrate on conceptual studies of high-power mucrowave generators in connection with slow-wave structures.
Konstantin G. Kostov was born in Sofia, Bulgaria,
on February 5, 1962. He received the B.S. degree in
physics, the M.S. degree in plasma physics, and the
Ph.D. degree in plasma physics, from Sofia University, Sofia, Bulgaria, in 1984, 1986, and 1994, respectively.
In 1986, he joined the Plasma Electronics Laboratory, Sofia University, as a Research Assistant.
From 1995 to 1996, he was a Visiting Scientist at
the University of Brasilia, Bras Lia, Brazil. In July
1996, he was a Postdoctoral Fellow with Department
of Engineering Physics, McMaster University, Hamilton, ON, Canada. From
September, 1997 to September, 1998, he was a Postdoctoral Fellow with the
National Space Research Institute (INPE), São José dos Campos, Brazil.
Since 1999, he has been an Assistant Professor with Department of General
Physics, Sofia University. From 2000 to 2001, he was a Visiting Scientist at
the Associated Plasma Laboratory, National Institute for Space Research. His
current research interests include high-power microwave sources as gyrotrons,
monotrons and vircators, high-current electron beams, and plasma industrial
applications.