|"JOURNAL OF RADIOELECTRONICS" N 7, 1999|
NUMERICAL INVESTIGATIONS OF RELATIVISTIC BACKWARD WAVE OSCILLATORI.A.CHERNYAVSKY, V.M.PIKUNOV*
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
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
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  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 .
iii) One of the main merits of a traditional BWO is a very wide frequency range of tunability in it (more than one octave ), 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  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 , 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 E01 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 forE014 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  for Cherenkov type devices. The theory was successfully used before for the study of TWT [16,17], BWO  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 . 2) The real frequency and complex wavenumbers are used. The extended Floque's problem  is solved for finding eigenwavenumbers. 3) The using numerical methods based on the special directed orthogonalization algorithm with semi-inversion  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.
2. HYBRID WAVES AND WAVES WITH COMPLEX PROPAGATION CONSTANT IN SLOW-WAVE SYSTEM OF RELATIVISTIC BACKWARD WAVE OSCILLATOR
Let consider sinusoidal corrugated one-mode SWS with the next geometrical parameters, that were used in recent experimental and theoretical investigations : period of corrugation d=1.67 cm, its amplitude h=0.445 cm, an average radius of the SWS Rav=1.445 cm. An equation of such corrugated surface can be represented as R(z)=Rav+hcos(2pz/d), where z is an axial coordinate. The classical method of dispersion equation enables to investigate types and properties of eigenwaves of considered SWS and also to find propagation and attenuation frequency regions of such waves, to calculate phase and group velocities of waves, to determinate strength breakdown of eigenmodes. It is convenient to present dispersion diagram (DD) in two parts (Fig.2,3). The real part of DD Re(kzd/2p) determines the phase shift of the eigenwave over the period d of SWS (left columns on Fig.2,3). The imaginary part of DD 20lg(e)*Im (kzd/2p) is the magnitude of eigenwave attenuation over one period of SWS in dB (right columns). For a convenience only a finite region of a real part of dispersion diagram is shown here. This region includes branches for “0” and “-1” spatial harmonics only. Backward spatial harmonics have anomalous dispersion and its phase and group velocities have opposite directions. Corrugated SWS - a main part of BWO’s EDS - provides propagation of series of axisymmetric (Fig.2) and nonaxisymmetric (Fig.3) eigenwaves.
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 depthh (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 - E01 and HE11 - can propagate and can be in synchronism with electron beam in lowest passband. The cutoff frequency fc of HE11 mode lies lower than the one for E01 mode. But the p-cutoff frequency fp of HE11 mode lies above than the one for E01 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, E1n and H0n 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 E11 and H01 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 E01 modes branches intersection or by both E01 and E02 modes branches intersection, have properties of E01 mode on a main part of the curves and properties of E02 mode near the upper boundary of the transmission band. Let call this mode as a hybrid and denote it as E01-02.
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 E02-01 and H01-02 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 E01 and E02 modes forms the row of hybrid modes - E(R,C)01-02 and E02-01. Two of these hybrid waves E(R)01-02 and E02-01 can propagate and were observed experimentally . The hybrid mode E(C)01-02 has complex propagation constant and wasn’t observed in experiment . 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(E0n)=(1; 0) and s(H0n)=(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 H11 and E11 modes results in formation of several hybrid modes: HE11, 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  within 1%. Unlike of paper , 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  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-TE01 mode in notation ) has concavity near the p-cutoff. It differs from the calculated curve , which has convexity near the p-cutoff. This may be one of the reasons why DD near p-cutoff point cannot be measured .
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 Tsws=S(+)inside/S(+)in accordingly, where S(+)out, S(+)in and S(+)inside - values of electromagnetic power flow for the right running waves on output and input cross-section of EDS, and also in a middle cross-section inside the SWS correspondingly. Unlike the coefficient T, coefficient Tsws can be bigger than unit, that is due to existence inside the SWS an additional power flow of waves reflected from both its ends. The appearance of such waves inside the SWS connects with unmatchings on both ends of corrugated waveguide piece. It leads to standing waves formation that localized only inside the SWS. An existence of such an additional flow of reflected waves doesn’t display on EDS output and in the transmission coefficient T correspondingly. Therefore, using jointly with a usual transmission coefficient T an additional coefficient Tsws, we can get an additional information about the processes that occur inside of the open resonator.
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 Lsws=6d, which joints with input and output pieces of a smooth waveguide with the radius R (see illustration on Fig.4a). The parameters of such corrugated SWS are the same as used in the experimental investigations : maximal radius R=1.91 cm, period of corrugation d=1.67 cm, amplitude of corrugation h=0.41 cm. Real boundary conditions on the ends of this EDS were modeled by the radiation conditions in the smooth waveguides. The model enables to study resonance properties of piece of corrugated waveguide, as an open resonator, and compare obtained theoretical results with an experimental  data. Let limit further our consideration by only an axisymmetric E01 mode as a mostly interesting for possible applications.
Fig. 4. a) Transmission coefficientsT (curve 1) and Tsws (2 – without, 3 – with losses consideration) versus the frequency for the corrugated SWS with N=6. Black circles here show the values of Q-factor for axial modes. b) Phase shift of E01 mode along the structure length versus a frequency.
In numerical investigation of electrodynamic properties of finite length EDS, it was excited by the smooth waveguide eigenmode E01 with some initial given value of power S(+)in. Frequency dependencies for transmission coefficients T (curve 1) and Tsws (2, 3) in this case are shown on Fig.4a in a frequency range, which corresponds to the SWS E01 mode transmission band. The indicated dependencies for both coefficients have sharply resonance nature. The value of normalized phase shift of corrugated structure eigenmode E01 along the SWS length Lsws is shown on Fig.4b, where f = kzLsws/p and kz – is the mode wavenumber. The N-1 maximums of transmission coefficient T appear on that frequencies inside the transmission band, where integer number of normalized phase shifts f = n (where n = 1,..,N-1) are realized; here N – is number of periods. It enables to introduce a numeration for such longitudinal resonances (or axial modes) of E01 mode by means an additional axial index n. Then it is convenient, the n-th longitudinal resonance of E01 mode to denote as E01n. By analogy it can be shown, that minimums of transmission coefficients are realized on the frequencies, where this wave has a phase shift along the SWS equals to f=(2n+1)/2 , where n = 0,...., N-1.
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 ). 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  measurement results. From the comparison of results presented here (Fig.4a) and the ones obtained before  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 Tsws 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 N-1 longitudinal resonances (maximums) inside the transmission band there are two additional resonance maximums on the frequency dependence of coefficient Tsws (see dotted curve 2 on Fig.4a). These resonances appear inside the SWS on the frequencies of lower and upper boundaries of E01 SWS mode transmission band with phase shifts f=0.0 (n=0) and f=6.0 (n = 6) accordingly. The existence of such type of resonances is in a good accordance with the experimental data , which obtained by means measurements of absorption resonance frequencies.
Fig. 5. The transmission coefficientTsws 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 (» 107). 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 Tsws 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 . 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 eigenmodesE01n : 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 Ez-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 (r = 1.0 cm), by the dotted lines - on the axis of the EDS (r = 0). The eigenmodes with axial numbers n=1,…,N-1 have n-1 axial variations inside the SWS (Fig.6b-f) and their axial distributions of Ez-field are like to the ones for the standing waves. An eigenmodes with numbers n=0 (Fig.6a) and n=6 (Fig.6g) are characterized by the amplitude attenuation from the input to output of the EDS. The longitudinal structure of own modes of EM-fields is weakly depended on the waveguide radius (compare solid and dotted curves on Fig.6), that is typical for the SWS of BWO. The presented axial distributions for Ez-field correlate with the experimental results  obtained for the closed resonator investigations. For comparison, additional field distributions, on the frequencies of two minimums of the transmission coefficient, are also shown on Fig.6h-i. They have another axial structure that is determined by a noninteger value of the phase shift f along the used SWS. Such distributions are also characterized by maximal amplitude on the EDS input.
3.2. Effect of horn antenna
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 . The transmission coefficient frequency dependence is determined by the smooth waveguide E01 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 Lsws=6d. The resonance properties of this open resonator are already determined by the finite length piece of corrugated waveguide and in fact are not differ from resonance curve presented on Fig.4a (curve 1). Using the calculated electromagnetic cross-fields distributions in the output of aperture of the horn antenna, we can obtain radiation pattern of the open resonator in a Kirchhoff approximation. For example, on the right part of Fig.1a the calculated radiation pattern on the frequency of an axial mode E014 is presented for the EDS that was used in experiment .
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 , 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 E01 mode. Parameters of the EDS are the next: length of SWS Lsws=6d, the cutoff neck with inner radius Rneck =1.15 cm and with complete length Lneck =3.1 cm. All results presented below were obtained taking into account of ohmical losses in EDS wall. It is also convenient to consider here all dependencies in logarithmic scale.
Fig.8. Transmission coefficientT 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 coefficientsT and TD versus a frequency for EDS with cutoff neck (solid curves) and without it (dotted). The open circles show values of Q-factor here. a), b) - for the distance D =0,625 cm; c), d) for the distance D =1.875 cm.
In the case, when the distance D > 0 it is convenient to use additional transmission and reflection coefficients, which defined as TD=S(+)D /S(+)in and RD=S(-)D /S(+)in correspondingly, where S(+)D and S(-)D - values of electromagnetic power flow for the right and the left running waves accordingly, calculated on a cross-section situated inside the gap between the cutoff neck and the SWS input in considered EDS. Using the transmission coefficient TD and reflection coefficient RD enable us to study processes inside the gap – for example, to get an information about the resonances formation inside it. For the distance D=0 dependence TD(f) is similar to the dependence T(f) which presented on Fig.8.
Fig.10. Transmission coefficientsTD and reflection coefficient RD versus a frequency for EDS with cutoff neck (D=1,875 cm) in the range of the lower (a), upper (b) resonances and inside the transmission band (c).
As the distance D increases (D > 0) all N resonances on dependencies T(f) and TD(f) are shifted further on a lower frequency region. It connects with an additional E01 mode phase shift a=kzD¤p along the distance D. For a certain value of the distance D the lower resonance lies below the cutoff frequency fc of SWS E01 eigenmode. In this case, the resonance is determined by a locked axial mode with a large value of Q-factor (see, for example, Fig.10), because of strong wave reflections from the cutoff neck and the SWS input. The locked resonance is localized exceptionally inside the gap (or a piece of smooth waveguide with radius R) situated between the cutoff neck and the SWS input in considered EDS and strongly attenuates out of the gap. For the considered value of the distance D (Fig.9a,b) there are exactly N-1 resonances inside the transmission band of SWS E01 eigenmode as it was found for used SWS (Fig.4a).
When the distance D increases furthermore a new locked resonance appears on the frequency range above the p -cutoff of E01 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 coefficientT 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 E01 eigenmode transmission band is limited on Fig.11a by two horizontal dashed lines – cutoff frequency fc and p -cutoff frequency fp of SWS E01 mode. The frequencies of using SWS axial modes are indicated inside the transmission band by the horizontal dotted lines which numerated by an axial index n. The dependencies of Q-factor values from the distance D are shown on Fig.11b for a few typical resonances that are marked on a both pictures by the same capital letters A¸G.
The character of all indicated changes is determined by the value of E01 mode phase shift a=kzD¤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 D changes on a value » l/2 all resonances substitute the next ones and new additional resonance appear above the transmission band. Here l - is wavelength of SWS (for D << l/2 ) or of a smooth waveguide (for D ³ l/2 ) E01 eigenmode. It is convenient all new resonances, that appeared only with the cutoff neck presence, to number by an additional index m (m=0,1,2,….; see Fig.11a), which is closely connected with phase shift a and can be defined as m=2D/l. From the obtained results we can also conclude, that for the certain value of the distance D, the N+m resonances on the frequency dependence of the transmission coefficients are realized.
For a some fixing frequency range it is also useful to distinguish two extreme types of the phase shifts, which are equal to a=(2l+1)/2 and to a=l accordingly (l = 0,1,2,….). These two types of phase shifts (and values of the distance D correspondingly) strongly differ from each other by the frequencies and Q-factors of axial mode resonances, that are especially noticeable inside the transmission band. In the cases of phase shift a=(2l+1)/2 there are only N-1 resonances inside the transmission band and all their frequencies practically coincide with the ones for the EDS without cutoff neck (which are indicated on Fig.11a by a dotted lines). For such values of the distance D the gap plays role of quarter-wave piece, which doesn’t influence on the resonance properties of the used SWS. If phase shift a=l there are N resonances inside the transmission band and all their frequencies are intermediate in respect to the ones for the EDS without cutoff neck. For such values of the distance D the cutoff neck influence strongly changes the resonance properties of the SWS inside the considered passband and the resonance Q-factor approaches here its local maximal value.
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 , 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 . The waves have properties like both an electromagnetic field and electron beam . 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 B0=2.2 T. The electron beam parameters are: radiuses Rb=1.0-1.1 cm, voltage Ud=560 kV and current Ib=5.25 kA. Using SWS has next parameters: period of corrugation d=1.48 cm, its amplitude h=0.225 cm, an average radius Rav=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 , 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 indexn=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 HE11 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 E01 and HE11 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].
5. RESONANCE MECHANISM OF GENERATION FREQUENCY FIXATION
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 Ist and Lst correspondingly.
As it was found  for electron beam by voltage on diode Ud=560 kV, with current Ib=5,3 kA and beam radiuses Rb=1.0¸1.1 cm the microwave generation can occur, when the SWS in the device has a length Lsws ³ Lst=5d (d=1,48 cm). In a case of Lsws=5d the frequency of generation was found equal to 9.65 GHz. These results are in a good accordance with an experimental data .
Fig.15. An amplification coefficientG 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 Lsws=6d was investigated numerically in detail. This SWS has analogous resonance properties as it has one considered above (Fig.4), but differs from it by a significantly modest values of axial modes Q-factor, which are in a range 40¸120 . The part of transmission coefficient frequency dependence is shown on Fig.15 by a dashed curve near the p -cutoff of the operating E01 mode. The amplification frequency dependencies in the same range are shown here for the three different values of voltage Ud and for the corresponding values of electron beam starting current Ist too. When value of voltage varies, the generation frequency varies too and can occur in the different part of the transmission coefficient frequency dependence.
Fig.16. Frequency (curve 1) and starting currentIst (2) of microwave generation for BWO based on SWS with Lsws=6d versus the voltage. Frequency dependency of transmission coefficient T for using SWS is shown here too by a dashed curve 3.
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 Ud > 500 kV it tunes near the resonance frequency of axial mode E014 and the range of tunability is approximately 3%. The obtained type of tunability and its range are in a good accordance with the analogous experimental data . The change of starting current has some features, which closely correlate with the resonance properties of using SWS: the minimal values of starting current occur on the voltage for which BWO generation frequency corresponds to maximums of transmission coefficient for considered SWS and vice versa.
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 Lsws 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 Ud=560 kV) are presented on Fig.17, where all resonances are marked by the same numbers 1¸ 3. For the comparison the value of operating electron beam current Ib is shown here by a dotted line. When the SWS length Lsws ³ 10¸ 14d starting current for the next order resonance became less than operating value of electron beam current (Ist < Ib) and the device excitation on the next order resonance frequencies becomes possible too. For such values of SWS length the problem of axial mode competition is a very actual and can also results in these modes’ switching processes .
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 Lsws=8d (d=1.48 cm), that was in detail experimentally investigated [3,25]. As it was found numerically, the starting current for such SWS has value » 1/5Ib (see Fig.17b). The EDS based on such SWS and with the cutoff neck has all the same quasi- periodical resonance properties, as they were for one considered above (Fig.11). Therefore we can suppose, that when the relativistic electron beam loads such EDS, its resonance properties quasi-periodically depend from the distance D too.
Fig.18. The values of frequencies and starting current of generation in relativistic BWO versus the distanceD.
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 E01 mode and is in a good accordance with the experimental data .
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 Lsws=6d (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 E01 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 coefficientT 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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Igor A.Chernyavsky, e-mail: firstname.lastname@example.org
Viktor M.Pikunov, e-mail: email@example.com