"ÆÓÐÍÀË ÐÀÄÈÎÝËÅÊÒÐÎÍÈÊÈ" "JOURNAL
OF RADIOELECTRONICS" N 7, 1999 |

NUMERICAL INVESTIGATIONS OF RELATIVISTIC BACKWARD WAVE OSCILLATOR

Laboratory of High Frequency Electronics, High Current Electronics Institute, 634055 Tomsk, Russia

*Department of Physics, M.V.Lomonosov Moscow State University, Vorobjevy Gory, 119899 Moscow, Russia]

Received July 7, 1999

CONTENTS

0. ABSTRACT

1. INTRODUCTION

2. HYBRID WAVES AND WAVES WITH COMPLEX PROPAGATION CONSTANT IN

SLOW-WAVE SYSTEM OF RELATIVISTIC BACKWARD WAVE OSCILLATOR

3. RESONANCE PROPERTIES OF OPEN RESONATOR

3.1 Axial modes of finite length corrugated waveguide

3.2 Effect of horn antenna

3.3 Influence of cathode cutoff neck on the resonance
properties of open resonator** **

4. ELECTRON WAVES, WHICH CAN BE EXCITING IN RELATIVISTIC

BACKWARD WAVE OSCILLATOR

5. RESONANCE MECHANISM OF GENERATION FREQUENCY
FIXATION** **

6. MATCHING OF CORRUGATED SLOW-WAVE STRUCTURE

7. REFERENCIES

Numerical investigations of electrodynamic systems (EDS) for experimental relativistic Backward Wave Oscillators (BWO) are made with help of computer codes based on linear theory methods. Dispersion diagrams for the nonaxisymmetric eigenwaves of the corrugated slow-wave structures (SWS) are calculated. Resonance properties of an experimental EDS containing finite length corrugated SWS, diaphragm cutoff neck and conical horn antenna, are investigated in detail. Frequency and starting current of generation are found for relativistic BWO based on such type of the EDS. The dependency of generation frequency with the electron beam voltage and cutoff neck location are investigated too. Obtained results are discussed and compared with known theoretical and experimental data.

**Keywords:** Relativistic Backward Wave Oscillator, Open Resonator, Corrugated
Slow-Wave Structure, Diaphragm Cutoff Neck, Conical Horn Antenna, Dispersion Diagram,
Longitudinal Resonance, Microwave Radiation Pattern, Nonaxisymmetric Modes.

It is known the BWO was a first successfully realized relativistic Cherenkov type device as a source of powerful microwave radiation [1,2]. Now intensive investigations are continuing [3-8] with the goal to improve approached characteristics of relativistic BWO and for its successfully application. The typical parameters in such devices are the next: an amplitude of accelerated voltage is » 0.3¸1.0 MV; voltage pulse duration is » 30¸350 ns; an operated electron beam current is » 1¸10 kA. But there exist a series of limitations that keep in a more widely using of relativistic BWO as a source of powerful microwave radiation:

i) Efficiency of such type devices come to 15¸ 25%.
Approached power level in 0,5¸3 GW is accomplished by a
significant radiation pulse shortening in comparison with the current pulse duration [3,7]. The effect of pulse shortening is conditioned by different breakdown
phenomena both on the surface and in a volume of EDS that connects with a high
concentration of microwave energy in the devices. One of the possible methods to increase
the pulse duration of microwave radiation in relativistic BWO is the attempt [9] of using an overmoded SWS with **D**/l > 1, where **D** - is maximum of resonator
diameter, l - radiation wavelength in a free space.

ii) Another important feature for using of these devices is needs to focus its output radiation. For this goal different types of transformers are worked out for axisymmetric electromagnetic modes transformation in a gaussian-type beam [6].

iii) One of the main merits of a traditional BWO is a very wide frequency range of tunability in it (more than one octave [10]), that approached by the electron beam energy changing. But experimental investigations [2,11] of relativistic BWO indicate on realization a significantly narrower frequency range of tunability £ 1¸3% in it, when a voltage is changed in a wide range of its value. Some attempts to expand the frequency range of tunability were made in a series of recent experimental investigations, for example, by help of variation of the gap value between the cutoff neck and SWS input [4] or using its partially plasma filling [5,8].

All above-mentioned features lead to necessarity to the further more detail both
experimental and theoretical investigations of relativistic BWO. A serious attempt to make
both experimental and theoretical investigations of open resonator loaded by corrugated
SWS were fulfilled during a last few years. The results of theoretical investigations of
characteristics of corrugated SWS [12], closed and open resonators for
relativistic BWO [13,14] were presented in a series
of recent publications. These data are a fine base for comparison and discussion of
different theoretical methods for one-mode BWO and TWT tubes. Typical open resonators,
that used in modern experiments for microwave generation in the relativistic BWO, usually
have an axisymmetrical EDS, which consist of diaphragm cutoff neck, piece of corrugated
SWS and a conical horn antenna [3,4,7,8] (Fig.1). Corrugated SWS is a main part of the
device EDS that provides propagation of synchronized with electron beam “-1” spatial
harmonic of operating E_{01} mode. Other nonsynchronized spatial harmonics also
produce a strong influence on the interaction with electron beam. Diaphragm cutoff neck on
the SWS input of the relativistic BWO is used for the microwave radiation reflection on
the device output. Then for the goal of an output radiation pattern formation the conical
horn antenna is used on the collector end of the BWO.

**Fig.1**. a) Schematic of relativistic BWO and radiation pattern for **E**_{014} axial mode of slow-wave structure.1 - cathode, 2 – magnetic system, 3 –
diaphragm cutoff neck, 4 – corrugated slow wave structure, 5 – conical horn antenna, 6
– an electron beam, 7- dielectric output window. b) Model of SWS used in numerical
investigations.

The main goal of this paper is making a detail investigation of electrodynamical
characteristics of open resonators in absence and with high-current electron beam in it.
Obtained results are compared with known experimental data [4,12-14,25] and discussed with other
theoretical predictions, if it is possible. All main parameters of SWS are closed to ones
used in experimental investigations [4,12-14,25]. The numerical investigations are made by the
methods of the linear theory [15] for Cherenkov type devices. The
theory was successfully used before for the study of TWT [16,17], BWO [18] and oversized BWO-TWT [19-21]
(or operating near the p -cutoff) type devices. Numerical
methods that used in the theory enable: to calculate dispersion diagrams of axisymmetric
and nonaxisymmetric SWS modes; to determinate transmission and reflection coefficients for
different types of overmoded (with ratio **D**/l £ 15¸ 25) open
TWT-BWO resonator (Fig.1); to calculate
radiation pattern; to find electromagnetic fields along the length and in any
cross-section of open resonator; to detect starting currents and generation frequencies of
operating or parasitical modes and so on.

The important features of the using theory are the next: 1) An electrodynamical description of electromagnetic fields are realized by the incomplete Galerkin’s method in which projective system contains eigenfunctions of Dirichlet's and Neyman's problems for circular membrane. Coordinate system contains also additional basis functions that are orthogonal to projective basis functions and have amplitudes determined by boundary conditions. This method permits take into account exact boundary conditions, that prevent Gibss phenomena and improve converging of Galerkin's rows [22]. 2) The real frequency and complex wavenumbers are used. The extended Floque's problem [23] is solved for finding eigenwavenumbers. 3) The using numerical methods based on the special directed orthogonalization algorithm with semi-inversion [15] enables to study of finite length overmoded SWS, that in particular permits to calculate an electromagnetic fields of overmoded conical horn antenna. Then the pattern for the BWO radiation is calculated by using the obtained field distributions in the output aperture of the antenna. 4) 3D linear model of electron beam and electromagnetic fields interaction is used.

Let consider sinusoidal corrugated one-mode SWS with the next geometrical parameters,
that were used in recent experimental and theoretical investigations [12]:
period of corrugation **d**=1.67 cm, its amplitude **h**=0.445
cm, an average radius of the SWS **R _{av}**=1.445 cm. An
equation of such corrugated surface can be represented as

**Fig.2**. Real and imaginary parts of dispersion diagrams for lowest
axisymmetric eigenmodes of the SWS (upper diagrams) and creation of slow waves within an
increasing of corrugation depth **h** (low diagrams).

The dispersion curves for waves (Fig.2,
3) show, that there can
exist electromagnetic waves with pure real, pure imaginary and a complex wavenumbers. The
modes with pure real wavenumbers are propagating waves in their transmission bands. For
one-mode SWS (D/l
» 1) only two waves - E_{01} and HE_{11} - can propagate
and can be in synchronism with electron beam in lowest passband. The cutoff frequency **f**_{c} of HE_{11} mode lies
lower than the one for E_{01} mode. But the p-cutoff frequency **f**_{p}** **of HE_{11}
mode lies above than the one for E_{01} mode. The modes with
pure imaginary wavenumbers are attenuated waves, that realized in frequencies lie below of
a lowest boundary of corresponding passbands. The waves with complex wavenumbers exist on
frequencies lie above the upper boundary of corresponding passbands. When an electron beam
loads SWS structure, the waves with imaginary and complex propagated constant to
participate in creating of wake fields of electron bunches. In spite of a lot amount of
papers devoted to investigations of corrugated SWS there is absent a satisfactory
classification even for axisymmetrical types of its modes till now. One of a useful method
of classification is to study a behavior of dispersion curves, when corrugation amplitude **h** is changed from its zero value to normal one. For this purpose,
we at first calculate DD (see low diagrams on Fig.2, 3) for smooth waveguide (**h**=0
cm - dotted curves) and for small depth of corrugation (**h**=0.01
cm - solid curves). It must be mentioned, that the usual classification based on a number
of the modes fields variations along waveguide radius is inapplicable here, because in the
one-mode waveguides high type waves have a hybrid nature. The properties of such waves can
significantly changed inside its transmission band; it concerns to the number of fields
variations too (see low diagrams on Fig.2,
3).

Note that in a smooth waveguide, E_{1n} and H_{0n} modes have degeneracy for all cross-indexes **n
**and its dispersion diagrams are coinciding. But in corrugated waveguide they
have different dispersion diagrams, see for example dispersion branches E_{11}
and H_{01} on Fig.2, 3.
In the case of small amplitude of corrugation, we can speak about an interaction of smooth
waveguide modes on the SWS discontinuities. For the case of axisymmetric waves (with
azimuthal index n=0) an interaction
between the TM and TE modes is absent. Directly from the dispersion diagrams one can see,
that on the points of dispersion curves intersections, the regions with complex wavenumber
are formed. For small amplitude of corrugation **h,** hybrid
modes are formed by intersection of two branches. If amplitude of corrugation is large (**h** =0.445 cm, upper row on Fig.2, 3), the pair of hybrid modes with real and complex
propagation constant are already formed. For example, dispersion curves, that is defined
by the E_{01} modes branches intersection or by both E_{01} and E_{02} modes branches
intersection, have properties of E_{01} mode on a main part
of the curves and properties of E_{02} mode near the upper
boundary of the transmission band. Let call this mode as a hybrid and denote it as E_{01-02}.

**Fig.3**. Real and imaginary parts of dispersion diagrams for lowest
nonaxisymmetric eigenmodes in the SWS (upper row). Slow waves formation pictures (low
row).

By analogy we can consider the formation of E_{02-01} and
H_{01-02} hybrid modes. As the value of amplitude **h** increases an imaginary part of wavenumber of complex modes
increases too; the frequency regions of attenuation are expanded. For considerable values
of ratio **h**/l hybrid modes are split on a two: one of them stays with a real
wavenumber, but another already has a complex wavenumber (see the upper row of dispersion
diagram on Fig.2). Analysis of Fig.2 permits to
conclude, that interaction of the E_{01} and
E_{02} modes forms the row of hybrid
modes - E^{(R,C)}_{01-02} and E_{02-01}.**
**Two of these hybrid waves E^{(R)}_{01-02} and E_{02-01} can propagate and were observed experimentally [13]. The hybrid mode E^{(C)}_{01-02}
has complex propagation constant and wasn’t observed in experiment [12].
That kind of wave may be detected only in special experiment, because they have strong
attenuation. The complex mode may play important role in interaction with bunched electron
beam.

A characteristic feature of nonaxisymmetric modes is an interaction both the same types
of waves and different TM- and TE- types of waves, that leads to the complication of whole
picture of interaction. An electromagnetic field of such waves contains all six nonzero
components. For this waves classification, let consider a vector value s = (S^{(E)};S^{(H)})/(çS^{(E)}ç+çS^{(H)}ç), where S^{(E)} and S^{(H)} - the electromagnetic power flux (or the Poynting vector), which are
transferred by an electrical and a magnetic part of mode correspondingly. For example, in
the case of azimuthal index **n**=0, s(E_{0n})=(1; 0) and **s**(H_{0n})=(0; 1) in any cross-section
accordingly. For the case of nonaxisymmetric modes both components of the vector s are as a rule nonzero and can have different meanings in different
points of dispersion curves of the mode. The interaction of H_{11} and
E_{11} modes results in formation of several hybrid modes: HE_{11,} EH^{(D,R)}_{11}, EH^{(D,C)}_{11},
EH^{(U)}_{11}. One of such modes, that have
lower transmission band, we denoted here by upper index (D); another
mode we denoted by upper index (U). For the cases of a big value of amplitude **h** the lower mode is split on two ones – with a real and a complex
wavenumber, that supplied by additional indexes (R) and (C) accordingly. It must be
mentioned, ever more complicated cases of waves coupling may be exist.

The dispersion diagrams for waves with real propagated constant and azimuthal indexes n=0,1 showed on Fig.2,3 coincide with measured ones [12]
within 1%. Unlike of paper [12], we consider here an imaginary part of
wavenumber, which is complex in a whole case. It enables us to obtain an additional
information about properties of SWS eigenwaves. Note, that used in [12]
mode classification differs from using one by us and has more specialized field of
application. Our study also shows that dispersion curve for EH^{(R)}_{11} mode (quasi-TE_{01} mode in notation [12]) has concavity near the p-cutoff. It differs
from the calculated curve [12], which has convexity near the p-cutoff. This
may be one of the reasons why DD near p-cutoff point cannot be
measured [12].

3. RESONANCE PROPERTIES OF OPEN RESONATOR

The resonance properties of considered SWS can be investigated by studying of power
transmission coefficient calculated both on output of EDS and inside the used SWS. The
first is defined as **T=S ^{(+)}_{out}/S^{(+)}_{in}**
and the second as

3.1. Axial modes of finite length corrugated waveguide

To learn main features of wave propagation, we used a brief model of the EDS, that
contains a piece of corrugated waveguide with length equal to **L _{sws}**=6

**Fig. 4**. a) Transmission coefficients **T** (curve
1) and **T _{sws} **(2 – without, 3 – with losses
consideration) versus the frequency for the corrugated SWS with

In numerical investigation of electrodynamic properties of finite length EDS, it was
excited by the smooth waveguide eigenmode E_{01} with some initial given value of power **S ^{(+)}_{in}**.
Frequency dependencies for transmission coefficients

We can also mention, that an amplitude of transmission coefficient **T**
on all resonance frequencies reaches unit, that characterizes a complete transmission of a
given on input an initial power flow and a practically complete absence of its reflection
from SWS. At the same time, the amplitude of the transmission coefficient **T**
between its resonances doesn’t drop to zero, as it is in a case of a closed resonator
(see in [13]). It is due to a difference in a nature of power losses:
on a radiation - for open and on an own (ohmical) - for closed, resonators.

**Q**-factors for all longitudinal resonances of coefficient **T,**
which are situated inside the lowest transmission band, are shown on Fig.4a
by means the black circles (right axis). These **Q**-factors were
calculated as the ratio of value of central resonance frequency to resonance width on
semi-height of its amplitude. The maximums of **Q**-factor are
reached near to the transmission band boundary and are equal to values ³
400.The resonances lying in a center of the transmission band have a minimal value of the **Q**-factor » 120 - 150. The calculated
values of **Q**-factor and its dependence from frequency for the
considered SWS are in a good accordance with obtained in [13]
measurement results. From the comparison of results presented here (Fig.4a)
and the ones obtained before [18] we can conclude that when amplitude
of corrugation decreases, the values of longitudinal resonance Q-factors are reduced too.

Unlike the transmission coefficient **T**, amplitude of
transmission coefficient **T _{sws} **on all resonance
frequencies inside the transmission band reaches values noticeably higher than unit. It
can be explained by an existence of an additional power flow of reflected waves inside the
SWS from both its unmatched ends. Except the indicated

**Fig. 5**. The transmission coefficient **T _{sws }**versus a frequency near the upper boundary of the transmission band without (1)
and with (2-4) ohmical losses: 2 – copper, 3 – steel, 4 – graphite.

The calculated values of **Q**-factors for resonances, which are situated on the
transmission band boundaries, didn’t show on Fig.4a, because were
too large (» 10^{7}). The curve 2 on Fig.4a
was obtained without ohmical losses in open resonator walls that result in eigenvectors
degradation on the frequencies corresponding to both boundaries of the transmission band.
To obtain a more correct result for these types of resonances, the ohmical losses must be
took into account, that leads to disappearing of the eigenvectors degradation. The
dependence for coefficient **T _{sws} **obtained with accounting of losses is
shown by a solid curve 3 on Fig.4a. The curve 3 in fact coincides
with curve 2 inside the transmission band and significantly differs from it on the both
boundaries of the passband.

More detail comparison of results obtained for the resonance situated on an upper
boundary of the transmission band (or p -cutoff) in the cases
without and with ohmical losses for a few type of waveguide materials are presented on Fig.5. The accounting of losses results in a significant reduction of
the resonance amplitude and its **Q**-factor, which still have
reasonable values. For example, in a case of copper waveguide wall **Q**-factor is
equal » 3000, but for the steel it is already equal to » 1000. These values of **Q**-factor are in
a good accordance with an experimental data [13]. Then in cases of
materials with a bigger value of wall resistance the resonance practically disappears
(see, for example, curves 4 on Fig.5 for graphite waveguide wall). It
must be also mentioned that in the case of the resonance situated on a lower boundary of
the transmission band calculated results are very similar to ones presented on Fig.5.

**Fig.6**. Normalized longitudinal distributions of axial electrical
field amplitudes of eigenmodes E_{01n}
: a)-g) – on the frequencies of transmission coefficient maximums;
h),i) - on the frequencies of its minimums

Let now consider in detail longitudinal distributions of normalized **E _{z}**-fields
for a different axial eigenmodes of the corrugated SWS. Such distributions calculated
without losses consideration are presented on Fig.6: by the solid
lines - on the fixed radius near the SWS surface (

An important element of open resonator, which used in a relativistic BWO and TWT
experiments, is a conical horn antenna (Fig.1a). It joints with the
SWS output and using for microwave radiation. On Fig.7a calculated
transmission coefficient is presented for the open resonator, that contains a piece of a
smooth waveguide connected with the conical horn antenna (see illustration). The sizes of
output system are the same as the experimental dimensions [13]. The
transmission coefficient frequency dependence is determined by the smooth waveguide E_{01}
mode dispersion and by the horn antenna reflections.

**Fig.7**. Transmission coefficient versus the frequency for an open
resonator without (a) and with (b) the slow wave structure.

On Fig.7b transmission coefficient is presented for the same open
resonator loaded by the corrugated SWS with the same length **L _{sws}**=6

3.3. Influence of cathode cutoff neck on the resonance properties of open resonator

The cathode cutoff neck provides practically complete reflection of operating backward
wave to the horn antenna direction (Fig.1a). As it was found [18], the resonance properties of such EDS depend on the value of the
distance **D** between the cutoff neck and SWS input (see Fig.1b). Let now consider the dependence **T**(**f**)
for such EDS excited from its input by E_{01} mode. Parameters of the EDS are the
next: length of SWS **L _{sws}**=6

**Fig.8**. Transmission coefficient **T**
versus a frequency for the EDS with cutoff neck (**D**=0 - solid curves) and without it (dotted curve). Open circles show values of **Q**-factor
(right axis).

For the case of distance **D**=0
calculated dependence **T**(**f**) is presented on Fig.8 by the
solid curve. For the comparison analogous dependence for used SWS without cutoff neck is
shown there by the dotted line. It is really, the existence of the cutoff neck results in
a sharply change of the obtained above dependence **T**(**f**) for used SWS. There
are **N** resonances on the frequency dependence for considered
EDS based on SWS, which containing **N**=6 corrugated periods. It can be explained by
the geometry of the used EDS. Actually, the periodical corrugated irregularities of SWS
have an inclination to the right side, that results in nonzero distance **D¢** between SWS and cutoff neck even for the case of distance **D**=0 (see Fig.1b).
An additional inclination of cutoff neck on an opposite side results in further increasing
of distance **D¢**. It shifts all
resonances frequencies to a lower region and leads to appearance of an additional
resonance near the upper boundary of the transmission band (Fig.8).
The presence of cutoff neck results in a significant changing of resonances **Q**-factor,
that is especially noticeable for two resonances located near both boundaries of the
transmission band.

**Fig.9**. Transmission coefficients **T **and **T _{D} **versus a
frequency for EDS with cutoff neck (solid curves) and without it (dotted). The open circles show values of

In the case, when the distance **D** >
0 it is convenient to use additional transmission and reflection coefficients, which
defined as **T _{D}=S^{(+)D
}/S^{(+)}_{in} **and

**Fig.10**. Transmission coefficients **T _{D}
**and reflection coefficient

As the distance **D** increases (**D**
> 0) all **N** resonances on dependencies **T**(**f**)
and **T _{D}**(

When the distance **D** increases furthermore a new
locked resonance appears on the frequency range above the p
-cutoff of **E**_{01} eigenmode (see, for example, Fig.9c,d).
At the same time, the lower resonance shifts down far from the transmission band, where
its amplitude on the dependence **T**(**f**) significantly decreases. It can be
explained that the resonance has completely locked mode nature, because connects with a
standing wave formed inside the gap and practically doesn’t transfer any power flow. It
means, that power flows of right and left running waves inside the gap are practically
equal to each other in the frequency region of the locked mode resonance (Fig.10a). Such resonances with high **Q**-factor cannot be observed
on the EDS output, because power flux is very small. When the power flows of right and
left running waves aren’t equal to each other, a combination of standing and running
waves inside the gap exists. In this case, such resonance can be observed, because has
noticeable amplitude on the EDS output (see Fig.9c,d and Fig.10b). It is also obviously, that inside the transmission band
exactly such situation is realized (Fig.10c).

**Fig.11**. a) The resonance frequencies of the transmission coefficient **T** versus the distance **D** (solid curves). Some typical resonances are marked here by help of capital
letters A¸G. b)
Values of **Q**-factor for the marked
resonances only versus the distance **D**.

For a certain value of the distance **D,** the upper
resonance lies inside the transmission band and all processes described above repeat
again. So we may suggest that the common view of the axial modes spectrum and the values
of their **Q**-factor can periodically repeat (especially inside the transmission
band), when the phase shift a changes on the value » 1. To study this problem it is convenient to consider the
dependencies of resonance frequencies versus the value of the distance **D,**
which are presented on Fig.11a. New resonances that appeared with
cutoff neck presence only can be numerated there by the index **m** (defined below).
Frequency range of the SWS E_{01} eigenmode transmission
band is limited on Fig.11a by two horizontal dashed lines – cutoff
frequency **f**_{c} and p
-cutoff frequency **f _{p} **of SWS

The character of all indicated changes is determined by the value of E_{01}
mode phase shift a=**k _{z}D¤p **inside the gap. The view of the axial mode spectrum
quasi-periodically repeats, when the phase shift a changes on
the value » 1. When the distance

For a some fixing frequency range it is also useful to distinguish two extreme types of
the phase shifts, which are equal to a=(2* l*+1)/2
and to a=

From the results presented on Fig.11b one can see, that the
behavior of **Q**-factor for different resonances certainty
determined by their location on the frequency region (see Fig.11a).
It permits to distinguish three types of resonances. First one includes resonances located
above the transmission band (for example resonances marked as **F** and **G**),
which have “humpbacked” dependency of **Q**-factor versus the distance **D**. The second one includes resonances, which locate inside the
transmission band and are characterized by the modest values of **Q**-factors (for
example resonances **C** and **D**). The third type of resonances are located below
the transmission band and characterized by the monotonous increasing of **Q**-factor
(for example resonances **A** and **B**).

Evidently, that all mentioned features of cutoff neck influence on the resonance properties of SWS, must be taking into account during the BWO designing and can strongly changes the device output parameters. All described here results help to explain an effect of quasi-periodical variations of generation frequency and efficiency of microwave radiation found in a recent experimental investigations [4,25] of relativistic BWO. The results presented here are very similar with the ones, obtained in investigations of two-stage overmoded open resonator [24], in which drift tube is equivalent to the gap situated between the cutoff neck and input of the SWS.

**Fig.12**. The reflection coefficient versus the frequency for EDS with
the cutoff neck (**D**=4,375
cm - solid curve) and without it (dotted). Black circles show values of **Q**-factor
here.

As calculation show, all resonances situated inside and not far from the transmission
band have a view of row of peaks on the frequency dependencies of the reflection
coefficient. By a certain values of the distance **D**
additional resonances may be observed experimentally near the both transmission band
boundaries and out of it (Fig.12). Therefore it can be
experimentally confirmed, that when the distance **D** is
varied, the number of observed resonances (especially - inside the transmission band)
periodically changed too accordingly to the results presented on Fig.11a.

4. ELECTRON WAVES, WHICH CAN BE EXCITING IN RELATIVISTIC BACKWARD WAVE OSCILLATOR

When a tubular high current electron beam propagates near the surface of the SWS its interaction with the SWS electromagnetic field results in its bunching. Then the bunched electron beam excites microwave electromagnetic fields. Such self-consistent process leads to the electron waves formation [26]. The waves have properties like both an electromagnetic field and electron beam [27]. They also transport both an electromagnetic and kinetic power flows at the same time.

**Fig.13**. Real and imaginary parts of dispersion diagram for a few
lowest axisymmetric (**n**=0) electron
waves, which are exciting in the corrugated SWS, which loaded by a tubular high-current
electron beam. Dispersion curves for the slow and fast waves of space charge (SWSC and
FWSC, accordingly), slow and fast cyclotron waves (SCW and FCW, accordingly) exciting in
the electron beam are shown here too.

The real and an imaginary parts of dispersion diagrams for the lowest axisymmetric
electron waves (**n**=0) are shown on Fig.13
for the case of finite value of axial magnetic field **B**_{0}=2.2 T. The
electron beam parameters are: radiuses **R**_{b}=1.0-1.1 cm, voltage **U**_{d}=560
kV and current **I**_{b}=5.25 kA. Using SWS has next parameters: period of
corrugation **d**=1.48 cm, its amplitude **h**=0.225 cm, an average radius **R _{av}**=1.425
cm. All parameters are closed to their experimental [3,25]
values. These electron waves include: E- and H-types of electromagnetic modes for using
SWS, slow and fast waves of space charge (SWSC and FWSC accordingly) and also slow and
fast cyclotron waves (SCW and FSW accordingly) of electron beam. An interaction of waves
of space charge (WSC) with the different branches of electromagnetic modes are occurred
near the frequencies of their intersections with latter's and can leads to the imaginary
part appearance on the dispersion diagram of electron waves. It results in the possibility
of exponential increasing (or decreasing) of electron waves along the SWS that corresponds
to the amplification (or absorption) of initial power flow.

It must be mentioned, that for the indicated value of magnetic field used in experiments [25], influence of an interaction of slow cyclotron wave with operating electromagnetic mode on the device operation is small and can be negligible, see Fig.13. We will suggest furthermore, that an electron beam propagate in the infinite magnetic field.

**Fig.14**. Real and imaginary parts of dispersion diagram for a few
lowest nonaxisymmetric electron waves with axial index **n**=1, which are exciting in the corrugated SWS loaded by a tubular high-current
electron beam. Dispersion curves for the slow and fast waves of space charge (SWSC and
FWSC accordingly) exciting in the electron beam are shown here too.

For the SWS with indicated parameters of corrugation, there is HE_{11} the
nonaxisymmetric electromagnetic mode with azimuthal index **n**=1
which lying in the operating frequency range, see Fig.14. The
comparison of Fig.13 and Fig.14 shows that in
the same operating frequency range the electron beam can effectively interact with E_{01}
and HE_{11} modes of using SWS. It can leads to the appearance of the problem of
the modes competition and can results in the realization of multiwaves operating regimes
in the considered device [28,29].

When an electron beam propagates near the surface of the corrugated SWS an
amplification of backward wave along the SWS can occur. An existence of unmatching on the
input end of using SWS makes the amplification of direct wave in this system possible too.
For the certain values of electron beam current (with fixed length of SWS) or for the
certain length of SWS (with fixed value of an electron current) the amplification can
sharply increases and there can occur microwave generation in the considered device. Such
values of electron current or SWS length can be called as starting and marked in below as **I _{st}**
and

As it was found [18] for electron beam by voltage on diode **U _{d}**=560
kV, with current

**Fig**.**15**. An amplification coefficient **G** versus the frequency for a three indicated values of voltage on diode. The
transmission coefficient **T** for the using unmatched SWS (see
its model on the picture above) is shown here by the dashed curve.

The device based on such SWS with length L** _{sws}=**6

**Fig.16**. Frequency (curve 1) and starting current **I _{st}** (2) of microwave generation for BWO based on SWS with

Lets now consider more detail how the frequency and starting current of generation in
such device change with the voltage variation. Calculated dependencies for frequency
(curve 1) and starting current (2) of generation versus the voltage on diode in a wide
range of its variation are presented on Fig.16. For the comparison
the frequency dependence for transmission coefficient **T** of considered SWS is shown
here by the dotted curve 3 too. When voltage increases, generation frequency increases
too. For the voltages **U _{d}** > 500 kV it tunes
near the resonance frequency of axial mode E

**Fig.17**. Frequency (a) and starting current (b) of generation versus
SWS number of periods for three order of resonances in relativistic BWO.

When the number of periods **N** of SWS increases the device starting current of
generation decreases and the frequency of generation shifts on a highest frequency region.
When the length **L _{sws}** is too big, there is a possibility of the device
exciting on the high orders of resonances. The obtained results for the frequency (a) and
starting current (b) of generation (for fixed voltage

As it was discussed above, the existence of cutoff neck leads to that the considered
EDS resonance properties strongly depend on the distance **D**.
An experimental investigations [3,25] show, that the
distance **D** variations results in a significantly and
quasi-periodically changes on the frequency and efficiency of microwave generation of the
relativistic BWO. Let now consider the SWS with the same parameters, but with the length **L _{sws}**=8

**Fig.18**. The values of frequencies and starting current of generation
in relativistic BWO versus the distance **D**.

The obtained results for the frequency and starting current of generation for the
considered device are shown on Fig.18. For the comparison the values
of frequency and starting current of generation for the using SWS without the cutoff neck
are indicated here by the dotted lines too. The calculated frequency and starting current
of generation complicatively and quasi-periodically depend on the distance **D**. The resonance properties of EDS loaded by a high current
electron beam can noticeably differ from the ones for “cold” EDS. The obtained here
dependencies correlate with corresponding experimental dependencies of frequency and
efficiency of microwave generation [3,25].

The radiation pattern calculated by the numerically founded cross-fields distributions
on the device aperture (with radius equal to 7.5 cm) is shown on right part of Fig.1a. The
pattern is determined by an operating **E**_{01} mode and is in a good
accordance with the experimental data [3].

6. MATCHING OF CORRUGATED SLOW-WAVE STRUCTURE

The "cold" resonance properties of using unmatched SWS strongly influence on
the frequencies and starting current of generation in relativistic BWO. Here we consider
the possibility of matching of using above corrugated SWS with **L**_{sws}=6**d
**(**d**=1.48 cm), for example, by two additional
irregularities with the same period **d** and with a linear decreasing amplitudes of
corrugation **h**, which are located on the both it ends (see
picture above the Fig.19). Frequency dependency for such SWS is
shown on Fig.19 by dashed curve near the p
-cutoff range of operating E_{01} mode. Used matching
results in the significant smoothing of resonance curve (compare with Fig.15).
It is evidently suggest that it must influence on the starting conditions of the generator
based on such SWS and on its properties of tunability with the voltage variations too.

**Fig**.**19**. An amplification coefficient **G** versus the
frequency for three indicated values of voltage on diode. The transmission coefficient **T** for the using matched SWS (see its model on the picture above) is shown here by
the dashed curve.

The frequency dependencies of amplification **G** are shown on Fig.19
for the same three values of voltage, as they were on Fig.15. The
using matching doesn’t expand the device range of tunability. It can be explained by
that the complete (an effective) length of matched SWS is noticeable bigger than unmatched
initial one. So the SWS whole length increasing compensates an increasing of starting
current with matching. Therefore such type of a simple and a very popular matching is
effective only for the “cold” system without an electron beam loading, see
dashed curve on Fig.19. But the resonance properties of matching SWS
loading by a high-current electron beam (“hot” system) are significantly different
from the ones for the SWS without electron beam. Therefore it is necessary to find any
other effective methods for matching such “hot” system.

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Authors:

Igor A.Chernyavsky, e-mail: tyger@lhfe.hcei.tsc.ru

Viktor M.Pikunov, e-mail: vmp@vmp.phys.msu.su