## 1. Introduction

Open resonators and open waveguides are widely used in millimeter and submillimeter wave electronics because they provide lower loss and higher Q-factor in comparison to the standard closed structures [Valitov et al., 1969; Shestopalov, 1985; Weinstein, 1966, 1995]. Examples of high performance measurement equipment employing open resonators (based on spherical or semispherical mirrors) include resonant wave meters, reference oscillators, systems for measurement of intrinsic electromagnetic properties of dielectric materials, and others [Valitov et al, 1969; Milovanov and Sobenin, 1980; Valitov and Makarenko, 1984]. Semispherical and spherocylindrical open resonators in combination with reflective diffraction gratings are used in various diffraction radiation oscillators [Shestopalov, 1976, 1985, 1991] providing higher frequency stability and output power in comparison to the standard devices such as traveling-wave tubes, klystrons, and magnetrons. Open resonators with echelette-type corner mirrors have been chosen as the basis for highly efficient Gunn and IMPATT diode oscillators. Quasi-optical resonators of such devices adopt reactive reflection and transmission-type schemes [Sukhoruchko et al., 2003]. Open resonators has found a wide practical application in relativistic electronics. Several types of oscillators and amplifiers have been created on their basis [Balakirev et al., 1993]. It has been demonstrated by [Weinstein and Solntsev, 1973] that Smith-Purcell effect (diffraction radiation) can be used to build an amplifier based on an open waveguide.

The constantly growing interest in the implementation of millimeter and submillimeter wave radiation in different areas of science and technology puts forward demands for components with high performance and flexible functionality. One of the most promising strategies for the development of such components is to modify their electromagnetic structure in order to increase operating frequency band and improve efficiency of interaction between the electron beam and electromagnetic wave. Following this strategy, several new approaches have been proposed based on modification of open coupled electromagnetic structures such as coupled open resonators [Shestopalov, 1991], open waveguides [Weinstein, 1995; Weinstein and Solntsev, 1973], open resonators with dispersion elements [Marshall et al., 1998], as well as the metal-dielectric structures [Shestopalov, 1991] which are particularly useful for electromagnetic wave excitation employing Cherenkov effect. Unfortunately, the practical realization of the proposed structures is a rather difficult task because of complicated electromagnetic analysis and a lack of systematic approach.

The objective of this chapter is to perform a comparative analysis of classical quasi-optical structures and their new modifications. The strategies for further development of these structures will be discussed based on the performed analysis.

The chapter starts with a description of basic properties of a classical regular open resonator as a basis for new modified millimeter and submillimeter wave coupled resonant structures. The properties of open resonators and open waveguides based on periodic metal and metal-dielectric discontinuities excited by both the electron beam and the surface wave of the dielectric waveguide are considered.

## 2. The coupled quasi-optical systems based on open resonators

This section is dedicated to the analysis of simple (regular) resonant systems and coupled quasi-optical systems based on periodic metal and metal-dielectric structures such as open resonators with diffraction grating, coupled open resonators and resonators with layered metal-dielectric structures.

### 2.1. The main properties of classical quasi-optical resonators

A classical quasi-optical resonator consists of two-mirrors. In the simplest case considered here, the open resonator contains two opposing flat infinitely thin parallel aligned disks. This system of mirrors is referred as plane-parallel resonator and known from optics as the main part of Fabry-Perot interferometer.

The plane-parallel resonators exhibit a number of valuable properties: sparse spectrum of resonant frequencies, homogeneous field along the symmetry axis of the resonator and the wavelength in the resonator is slightly different from the wavelength in the free space.

While simple, this arrangement is rarely used in practice due to the difficulty of alignment, comparatively large size, and insufficient mode separation. Therefore the resonators based on the reflectors with quadratic phase correction are more promising in the millimeter and submillimeter wave range. These type of resonators are referred as confocal resonators and contain spherical mirrors. Тhese resonators exhibit a better spectral resolution in comparison to the plane-parallel resonators. Besides, confocal resonators are less sensitive to misalignment. The resonator with spherical reflectors typically exhibits lower power loss per one propagation in comparison to the open resonator with plane mirrors having the same aperture. The other important advantage is the large separation between the fundamental and the higher order modes

where H is the distance between the mirrors;

Limiting the size of resonator's apertures results in radiation loss and has negligible effect on the field distribution in the open resonator. Therefore the field must be concentrated close to the center of the mirror in order to reduce the losses. This, in turn, restricts the choice of ratio between the radius of the curved mirrors and the distance between them. In order to construct resonators with the field concentrated close to the center of the mirror, the distance between the mirrors must be selected within the following intervals:

This expression is known as the condition of stability of the resonator with quadratic correction;

The behavior of oscillations in plane-parallel and spherical-mirror resonators is quite different. The field distribution in the plane-parallel resonator mostly depends on the dimensions of the plane plates, while field distribution in the resonator with spherical mirrors is mostly determined by their radius and the ratio of the distance between mirrors and the radius,

The semi-spherical resonators which consist of a plane and a spherical mirror have also received a great deal of interest in microwave and millimeter-wave applications. It is known that the fundamental modes of the semi-spherical resonator are represented by the azimuthal oscillations

### 2.2. Resonators with periodic metal grating

The plane-parallel mirror of the semi-spherical resonator can be substituted by a diffraction grating as it is shown in Fig. 1. Such an electrodynamic structure is often used in diffraction radiation oscillators - orotrons [Shestopalov, 1976, 1991; Marshall et al., 1998; Ginzburg et al., 2000; Bratman et al., 2002; Rusin et al., 2002].

The orotron's operation principle is based on the diffraction radiation effect caused by the electron beam propagating above the diffraction grating of the open resonator. The electron beam interacts with the incident field diffracted from the grating which results in oscillation and amplification of the electromagnetic signal. Therefore, the orotron's output characteristics are strictly defined by the properties of the implemented open resonator. The periodic structure in the open resonator of the orotron considerably changes the characteristics of the previously described classical resonant quasi-optical structures. The substitution of the plane mirror by a diffraction grating considerably increases the total loss resulting in the Q-factor degradation by almost four times. The decrease of the Q-factor occurs as the result of additional losses, which are originated from a power leakage of the waveguide waves propagating along the grooves to the edges of the mirror where the reflection coefficient is not equal to one.

To overcome this drawback, a semi-spherical resonator where only the central part of the plane mirror was covered with the diffraction grating, has been proposed [Shestopalov, 1976, 1991]. This resonator has a wider distance between the oscillation frequencies. The achieved radiation loss depends on the parameters and the position of the grating. The width of the grating defines the number of the oscillation modes excited in the open resonator and the frequency of the higher order resonances. Losses in the open resonator are greatly dependent on the ratio between the period of the grating and the wavelength. The maximally achieved Q-factor of the resonator also greatly depends on the groove depths of the reflective grating oscillations could be varied by several times.

The fundamental mode of the semi-spherical resonator with a local diffraction grating is

Corner-echelette open resonators are widely used for realization of semiconductor sources in the microwave and millimeter-wave range. For example, modifications of quasi-optical reflection and transmission-type solid-state pump oscillators with spherical-corner-echelette open resonator have been shown in [Belous et al., 2003]. As shown in [Sukhoruchko et al., 2003], the corner-echelette resonator has the following properties: the degree of sparseness of the spectrum is lower than for the resonator with plane echelette mirror; however, the spectrum contains the oscillation modes with extremely high Q-factor, which are known as the quasi-fundamental oscillation modes; the field of the quasi-fundamental oscillation modes is concentrated around the axis of the open resonator resulting in a larger power density in comparison to the fundamental and higher order oscillation modes; the field distribution close to the surface of the corner-echelette mirror transforms and near the center of the resonator becomes similar to the field in a rectangular waveguide; corner-echelette mirror can be considered as a multi-step impedance transformer.

### 2.3. Coupled open resonators

The work by [Shestopalov, 1991] is dedicated to the diffraction radiation devices employing coupled open resonators. The coupled resonators have an advantage of providing a wider operating frequency range in comparison to the single resonator structures. The coupling between open resonators can be realized either by means of the field diffracted at the edges of the mirrors using series positioning of the resonators (Fig. 2а) or the field diffracted on a metal-strip grating using parallel connection of open resonators (Fig. 2b) with respect to the axis of the distributed excitation source. In the electron devices, the electron beam is such a source. In case of experimental modeling of diffraction radiation it is the surface wave of the single-mode dielectric waveguide.

The system of series open resonators is, in the case shown in Fig. 2а, consists of two semi-spherical resonators with the common plane mirror realized as a reflective diffraction grating. In the parallel coupling case (Fig. 2b), a two-layer metal-strip diffraction grating is placed between the spherical mirrors.

Systems of coupled open resonators consist of spherical mirrors 1 with the radius R=60 mm and aperture A=55 mm reduced to 35 mm along the axis of the dielectric waveguide 2. The lower plane mirror 3 of the system shown in Fig. 2a is either a reflective or semitransparent diffraction grating and serves as a common mirror for the first and the second open resonator. In the system with parallel open resonators, plane mirrors 4 with semitransparent diffraction gratings in their central sections were placed between spherical mirrors 1.

Parameters of the gratings are chosen to ensure the operation at a frequency f0 = 46 GHz. These gratings transform the surface wave of the dielectric waveguide into a free space wave propagating normal to the surface of the grating [Shestopalov, 1976]. The energy is coupled out from the system through the coupling slots in the spherical mirrors. The signals are then fed to a detector and measured using a standard measurement equipment [Shestopalov, 1976, 1991].

The described coupled resonators have been analyzed with regard to their spectra and resonance characteristics of oscillation. The measured characteristics of the equivalent single hemispherical and spherical open resonators have been used as a reference.

Fig. 3 shows the resonant frequencies versus the distance between the mirrors (H) in the system with coupling through the diffraction field (Fig. 2a) and in a reference hemispherical open resonator. The data presented in Fig. 3 characterizes the capability of the considered resonance system to support a limited number of TEMmnq oscillation modes.

The data in Fig. 3a shows that for the hemispherical open resonator, the fundamental

Figure 3b shows the resonances of two coupled open resonators tuned to a frequency f=46 GHz. As can be seen from these spectra, the second hemispherical open resonator is excited at the edge points of the frequency band in the interval H=27–33 mm. There are no oscillations around the resonance frequency of the open resonator, which is due to the minimum amplitude of the diffraction field in a case when the diffraction-grating–dielectric-waveguide system emits radiation along the normal. Detuning from the frequency

Coupled open resonators with a strip grating at the center of the common plane mirror (Fig. 2b) exhibit similar properties. The decrease in the number of oscillation modes in such a system (Fig. 3c) is due to the selective properties of the employed diffraction grating [Shestopalov, 1991]: the intensity of radiation emitted from the volume of the open resonator to free space through the diffraction grating reaches its maximum at

The open resonator with spherical mirrors, which is a basis for the second scheme of coupled resonators (Fig. 2b) supports similar to the case of the hemispherical open resonators fundamental TEM00q modes. This follows from the analysis of the achieved resonance frequencies. The field distribution in an open resonator with spherical mirrors is the same as in the hemispherical open resonator [Shestopalov, 1976]. However, the distance between the resonance frequencies in the open resonator with spherical mirrors is two times smaller than in a hemispherical open resonator. Inserting an additional plane mirror with a strip diffraction grating in a spherical open resonator will result in the spectrum of the coupled system similar to the spectrum of the hemispherical open resonator (Fig. 3a). The metal-strip diffraction gratings couple two hemi-spherical open resonators simultaneously filtering out the angular spectrum of plane waves excited in the system. Consequently, the variation of the position of these diffraction gratings in the volume of the resonator with respect to the spherical mirrors changes the spatial distribution of the fields corresponding to the oscillation modes excited in the considered system of coupled open resonators. Similar to the hemispherical open resonator with a reflective diffraction grating, TEM20q and TEM02q oscillation modes, as well as the higher order oscillation modes arise due to introducing a coupling element such as a double-layer diffraction grating.

The measured data for the resonance curves of coupled open resonators indicates that the achieved bandwidth of the system becomes much broader when the open resonators are tuned to close frequencies rather than in the case when the resonators are coupled through the diffracted fields. Fig. 5 presents the response of the open resonators coupled through a strip diffraction grating and for the open resonator with spherical mirrors. The achieved bandwidth of the coupled system was observed within the range

Analysis of the achieved bandwidth

### 2.4. Open resonators with metal-dielectric structures

Coupled systems based on open resonators and open waveguides with metal-dielectric structures allow to realize different modes of energy transformation depending on parameters of the electromagnetic system [Shestopalov, 1991].

The simplest open resonator employing a metal-dielectric slab is shown in Fig. 6а. It consists of a metal plane and a dielectric slab with a planar metallic diffraction grating on its surface.

A more complicated case of the open resonator with a metal-dielectric structure is shown in Fig. 6b. The resonator consists of a spherical mirror, a plane mirror such as a reflecting diffraction grating, and a layered metal-dielectric structure between the two mirrors. Such an electromagnetic structure is often used in Cherenkov diffraction oscillators. Fig. 6b demonstrates possible modes of Cherenkov diffraction radiation excited by a source of electromagnetic energy distributed between the metal-dielectric grating and plane mirror.

The metal-dielectric slab (Fig. 6b) of the open resonator introduces qualitatively new electromagnetic properties in such a system. It is possible to attenuate the power in the open resonator, increase the amplitude of the oscillating wave and the value of Q-factor as well as to improve selectivity by choosing parameters of the metal-dielectric slab.

## 3. Coupled open waveguides employing periodic structures

This section describes the main properties of quasi-optical open waveguides with periodic metal-dielectric structures excited by distributed sources of electromagnetic energy such as electron beam or surface wave of a dielectric waveguide. Such structures are promising for the design of low-voltage amplifiers based on Smith-Purcell effect [Weinstein and Solntsev, 1973; Smith and Parcell, 1953] and other microwave and millimeter wave electron devices [Joe et al., 1994, 1997].

### 3.1. Amplifiers based on Smith-Purcell effect

Fig. 7 shows the structure of the amplifier using a planar layered metal-dielectric stack and based on Smith-Purcell effect. The open waveguide of the considered system consists of the periodic rectangular grating structure 1 with the period of 2l, width 2d and grating depth of h; the planar layered metal-dielectric structure 2 with the thickness

The electromagnetic problem is solved using the method of partial domains. The field in each domain is determined from the Maxwell equations, equation for the electron beam propagation, and corresponding boundary conditions. In order to obtain the dispersion equation, we have to perform the following operations: determine the linear approximation of the equation for the variable component of the convectional current intensity and the field in the beam, and transform it into a homogeneous form; determine the electromagnetic field in the interaction region in hot (with electron beam) and cold (without electron beam) regime.

To this end, the electric field E, the beam velocity

where

The following analysis of equation (3) concerns defining the propagation constant

where

The solution to the dispersion equation is found by using Newton iterative method for the range of electron velocities

The numerical analysis of the dispersion equation for an open waveguide with no dielectric layer shows that there are two direct waves and two waves traveling in the backward direction exist in the system without the presence of an electron beam. They have different wave numbers and corresponding phase velocities. In addition to the previously described electromagnetic waves in the open waveguide, there are also two electron beam waves: a fast space-charge wave and a slow space-charge wave. All four electromagnetic waves in the system while synchronized with the electron beam spatial waves have regions with a positive amplitude growth that allows for signal amplification and realization of the following regimes varying parameter

Introducing a dielectric layer with small values

The increase in permittivity

Parameter

It should also be noted that increasing the distance between the mirrors results in increase of a number of surface waves and decrease of the gain factor for the volume waves. In the extreme case when the values

### 3.2. Experimental modeling of coupled open waveguides

The experimental modeling is one of the most efficient methods for solving problems of diffraction electronics. The radiation of the electron beam is simulated by a surface wave in the planar dielectric waveguide placed above the diffraction grating. The modeling techniques have been sufficiently developed and summarized in the literature [Shestopalov, 1976, 1985, 1991]. Nevertheless, each structure has its own specific features which have to be taken into account while developing and realizing the experimental setup.

There are three components in the previously described electromagnetic system which can be considered separately during the experimental modeling of the wave processes in amplifiers based on Smith-Purcell effect. They determine the general electromagnetic properties of the open waveguide. These components are the dielectric waveguide which feeds the surface wave into the system; diffraction grating which transforms the surface wave from the dielectric waveguide into the volume wave; the planar layered metal-dielectric structure which serves for both a transformation of the surface wave into the volume wave for the dielectric layer and reflection of the radiation arriving from the diffraction grating - dielectric waveguide interface. Compared to the system without the metal-dielectric layer, the wave processes in the open waveguide with the metal-dielectric stack are more complicated in comparison to the systems without such a stack due to the presence and superposition of different waves such as the volume wave incident to the layered metal-dielectric structure from the diffraction-grating-dielectric-waveguide interface and the waves propagating in the dielectric.

The parameters of the diffraction-grating-dielectric-waveguide system are chosen to satisfy the condition of the volume wave existence in the open waveguide [Shestopalov, 1991]:

where

The period of the diffraction grating has been chosen such that the main lobe of the radiation pattern (

The distance between the dielectric waveguide and the surface of the diffraction grating, a, is a very important parameter for the optimization of the system. The diffraction of the surface waves on the diffraction grating is nontrivial in this case because the value a is chosen to be smaller than the wavelength. However, a strong coupling between the waveguide and the diffraction grating effects the field distribution in the waveguide and, consequently, the propagation constant βw. The strong coupling results in interference between the wave propagating along the waveguide and the wave being scattered by the diffraction grating. Such an interference might result in additional propagation modes in the waveguide and, consequently, in the parasitic spatial harmonics [Shestopalov, 1991].

The behavior of the planar metal-dielectric structure of the open waveguide is similar to the behavior of the shielded planar dielectric waveguide. In order to analyze the physical phenomena of the electromagnetic wave excitation in the layered metal-dielectric structure, the electromagnetic field can be represented as a composition of plane electromagnetic waves. Based on this, the metal-dielectric structure can support two types of waves: the one excited by the diffraction grating-dielectric waveguide interface (these waves not necessarily undergo total internal reflection in the dielectric for certain angles

The experiments were performed in the frequency range from 30 GHz to 37 GHz within the interval

Fig. 10 shows of the normalized radiation pattern in the open waveguide at the center frequency

For dielectric layers with

Fig. 10b (curve 1) demonstrates the patterns of the diffraction-grating-dielectric-waveguide radiating system. It is clear from the presented data that the main radiation maximum is in agreement with the calculated value of

Covering the dielectric layer with a metal (Fig. 7) results in the fact that the radiation arriving from the diffraction-grating-dielectric-waveguide system will be reflected and fed into the open waveguide volume exciting the wave along its axis. Correspondingly, there are two volume waves propagating in the system: the wave in the layered metal-dielectric structure and the wave in the volume of the open waveguide. These waves are coupled to each other by means of the surface wave of the common radiation source - the dielectric waveguide. The existence of the forward and backward coupled waves in the open waveguide might result in parasitic resonances during the modeling. The wave numbers are complex if there is a coupling between the direct and the backward waves. This indicates the excitation of complex decaying waves. The waves are synchronized and the power of the forward wave is pumped into the backward wave and vice versa. Such a power exchange is performed along the significant propagation distance if the coupling is weak. The propagation becomes impossible and the transmission line turns into a sort of a resonator for certain frequencies. In such a system the waveguide characteristics such as the standing wave ratio (SWR) and the transmission coefficient (Ktr = Poutput/Pinput, where Poutput and Pinput are the power values at the dielectric waveguide output and input respectively) become fundamental. The waveguide characteristics of the dielectric-waveguide-dielectric-layer system (curve 1), dielectric-waveguide-diffraction-grating-dielectric-layer system (curve 2) and the open waveguide system in general (curve 3) are represented in Fig. 11 for ∆ ≈ λ. The presented data indicates that the SWR of the open waveguide elements and the system in general are within the interval 1,05 1,4. These reflections are due to the out of band mismatch of the dielectric-waveguide-metallic-waveguide transitions. The achieved SWR is considerably different from SWR for the open waveguide with no dielectric layer which is approximately 2,0 (curve 4) due to the resonance nature of the system. Substantial changes in the behavior of the Ktr versus frequency are also observed. Curves 1 and 2 indicate an efficient transformation of surface waves into the volume waves, while graph 3 indicates the presence of the coupled waves in the system and it is substantially different from the behavior of Ktr for the open waveguide with no layered metal-dielectric structure in it (curve 4). It can be assumed that for ∆ ≈ λ a large amount of power escapes from the dielectric and propagates in the open waveguide. The observed maxima and minima of the spectrum of Ktr can be explained by the fact that the waves propagating in the open waveguide are combined in- and out of phase.

The increase in the thickness of the dielectric layer results in the fact that the most amount of power is concentrated in the dielectric which leads to decrease in coupling between the layered metal-dielectric structure and the dielectric waveguide, and, in general, increase in Ktr for the open waveguide components (Fig. 12, curves 1 and 2) at ∆ ≈ 4λ.

At the same time, the behavior of the transmission coefficient in the considered frequency band indicates the decrease in the coupling between the waves propagating in the open waveguide (Fig. 12, curve 3).

The analysis described above for the characteristics of the open waveguide and its components indicates that it is possible to control the electromagnetic processes in the system by varying the thickness of the dielectric layer: adjust the coupling between the radiation of the dielectric waveguide and the waves propagating in the open waveguide. The increase in coupling is useful for enhancing the efficiency of the interaction between the electron beam and the open waveguide fields in the amplifier applications. The decrease in the coupling is interesting for realization of power decoupling from the open waveguide through the dielectric layer.

## 4. The implementation of coupled quasi-optical systems in vacuum electron devices

A two-stage diffraction radiation oscillator has been realized using the structure shown in Fig. 2а in the frequency range

The device operates as an amplifier if the microwave signal is applied to the input of the first (with respect to the gun) resonator and the beam current J is less than the starting current

Figure 14 presents the diagrams of a vacuum electron devices with open resonators connected in series with respect to the axis of the electron beam. An orotron shown in Fig. 14a consists of two coupled open resonators 1. Each of these resonators consists of two mirrors 2 and 3. Energy is coupled out through a waveguide in mirror 2. Mirror 3 has a parabolic cylinder shape. Metal-strip diffraction gratings 4 located in the center of the adjacent parabolic mirrors 3 are made of metal bars. The electron gun 5 generates a focused electron beam 6 and is placed between the parabolic mirrors 3. A collector 7 is positioned at the end of the interaction region.

The operation of the orotron can be described in the following way: the electron gun generates a focused electron beam which than experiences a bunching within the small interaction length due to the spatial charge in the interaction zone formed by the open resonators and gratings. The diffraction radiation is produced in the open resonators as electrons propagate through the gap between the diffraction gratings. The electrons are than striking a collector at the other end of the interaction region. The orotron operates as an oscillator if the electron beam current is much higher than the starting current. The orotron operates as an amplifier if the condition of self-excitation is not satisfied and a signal from an external microwave source is fed to the input of one of the resonators. It should also be noted that the orotron may function as a frequency multiplier if using two coupled open resonators. This device is a low-power oscillator. The increase of the electron beam current density is limited due to overheating of the strip diffraction grating.

A higher power level can be achieved in diffraction radiation oscillators based on coupled open resonators schematically shown in Fig. 14b. The design and the principle of operation of such a device are similar to the design and the principle of operation of the previously described orotron. The coupling of resonators 1 is achieved through the slots in the identical reflective diffraction gratings 4 placed in the center of the adjacent mirrors 3 and perpendicularly oriented with regard to the planes of these mirrors. The electron beam is focused with a magnetic field. The use of bulky gratings attached to the mirrors simplifies the temperature dissipation and, consequently, allows for higher electron beam currents. Furthermore, one of the resonators in such a system may be realized with an option for mechanical tuning where a moving short-circuit plunger located on the opposite side of the coupling slot. Figure 15 shows the oscillation bandwidth and frequency tuning characteristic for different distances h of the plunger for the case when the open resonator is centered at f0 = 36 GHz.

The presented data shows that, one can smoothly tune the oscillation frequency within a sufficiently broad frequency range by mechanically tuning the volume resonator with a fixed value of H for the mirrors in the open resonator. The variation of the output power in the considered frequency band does not exceed 3 dB. This characteristic of the considered device indicates the possibility for improving the vibration stability of the system in comparison to the vibration stability of systems with mechanical tuning of mirrors. The grating-coupled open resonators could also be used to build reflection type diffraction radiation oscillators [Shestopalov, 1991]. In this case, the collector should be replaced by an electron reflector, producing a backward electron beam. Such devices exhibit low starting currents and able to operate in the regime of stochastic oscillations [Korneenkov et al., 1982].

The wide functionality of open resonators with layered metal-dielectric structures allowed to build several types of diffraction based devices with complex resonant structures such as Cherenkov diffraction oscillator and Cherenkov backward-wave tube. Fig. 16 shows the example of Cherenkov backward-wave tube and Cherenkov diffraction oscillator.

The electron beam 1 of the backward-wave tube is generated by the electron gun 2. The beam propagates through the channel 3 formed by the adjacent surfaces of the resonator 4 to the slow-wave structure 5. The electron beam interacts with the field of the slow-wave structure 5 resulting in modulation of charge density. Simultaneously, Cherenkov radiation occurs when the electrons velocity exceed the phase velocity of the electromagnetic wave in the dielectric. The radiation is directed into the dielectric. The resonator 4 has a field distribution allowing a feedback (solid lines with arrows). Oscillations occur in the resonator effectively extracting power from the modulated electron beam via the strip grating 6 when the frequency is synchronized with the eigen frequency of the resonator. The power is coupled from the resonator 4 via the waveguide 7 with ε1 > ε. The synchronization between the electron beam and the wave in the dielectric is achieved by choosing the proper value for ε and adjusting the accelerating voltage for the electron beam.

Fig 16. Realization of Cherenkov backward-wave tube and Cherenkov diffraction oscillatorThe similar electron optics is used for excitation of Cherenkov diffraction radiation. The slow-wave structure (diffraction grating) 5 is positioned in the central part of the fixed mirror. The moving mirror 8 with a coupling slot 9 is used for coupling the power out of the device. In contrast to the backward-wave oscillator, the geometrical parameters of the gratings 5, 6 are optimized for efficient excitation of radiation in the normal direction with respect to the axis of the electron gun 2 (dotted oscillation pattern in Fig. 16) and for maximum power density of Cherenkov radiation within the dielectric resonator 4.

Recently, significant attention is drawn to amplifiers based on Smith-Purcell effect, which has been described in section 3. An amplifier employing sheet electron beam is shown in Figure 17.

The open waveguide with length L is formed by the surfaces of parallel passive 1 and active 2 mirrors realized as reflecting diffraction gratings with the periods

where

The range of angle

The prototype of the suggested travelling wave tube has been realized in the V band. The open waveguide was formed by two cylinder-shaped mirrors (the passive mirror with the curve radius

A further improvement of the output parameters of the amplifier could be achieved by increasing the interaction region and the electron beam current. This could be achieved, for instance, by means of using axial-symmetric electromagnetic systems and a better electron focusing optics.

## 5. Conclusion

The chapter provides a summary of results on both the classical quasi-optical systems forming a basis for development of new modifications of oscillation systems of the microwave and millimeter-wave band devices and more advanced coupled electromagnetic systems with complex periodic structures such as coupled open resonators, open resonators and waveguides with layered metal-dielectric structures. It is demonstrated that the coupled open resonators exhibit wider frequency tuning range while preserving high values of Q-factor. Coupled systems such as open resonators and waveguides with layered metal-dielectric structures have qualitatively new properties: by varying the parameters of metal-dielectric structure one could achieve attenuation or amplification of the oscillations and their selection. New modifications of Cherenkov traveling wave tube such as Cherenkov diffraction oscillator and amplifier based on the Smith-Purcell effect are suggested and realized.

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