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 topic 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 topic 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.
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.
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. These 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 TEMmnq , where m, n = 0,1, 2,… is the number of half-waves in transverse direction and q is the longitudinal index which corresponds to the number of half-waves in the direction of propagation. For the resonator with spherical mirrors the resonance distances or the resonance wavelengths of the oscillation modes should comply with the following condition:
where H is the distance between the mirrors; X is the wavelength in the open resonator;
are the curvature radii of 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; g2 are the parameters that depend on geometry of the resonator. 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
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 TEMm0q . If the field spot on the plane mirror is considerably smaller than its diameter then the semi-spherical resonators can be substituted by the equivalent resonators with two spherical mirrors having doubled distance between them. The distribution of amplitudes in both cases is identical to a high degree of accuracy. The Q-factor of the semi-spherical resonator depends on diffraction losses at the edges of the plane and the spherical mirrors, ohmic losses in the mirrors, the coupling losses and the losses related to attenuation in the medium.
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.
Fig. 1. Semispherical open resonator with diffraction grating
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 TEM20q . The research in [Shestopalov, 1976, 1991] proved that the perturbation caused by the grating is insignificant in such a system if the minimum of the field distribution is above the boundary between the grating and the mirror. This is the case when the width of the diffraction grating is larger or equal than the width of the main lobe in TEM20q oscillation mode.
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.
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. 2a) 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.
Fig. 2. Electrodynamic systems based on coupled open resonators: a – series connection of open resonators; b – parallel connection of open resonators
The system of series open resonators is, in the case shown in Fig. 2a, 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.
Fig. 3. Spectra of resonant frequencies of (a) a hemispherical resonator and (b, c) a diffraction-coupled resonators with (b) reflective and (c) strip diffraction gratings.
The data in Fig. 3a shows that for the hemispherical open resonator, the fundamental TEM00q modes exist in the entire frequency range f = 45 – 47 GHz while changing the distance between the mirrors. The implementation of a dispersive element such as the reflective diffraction grating in the open resonator allows for the modes with transverse indices m and n. The TEM20q oscillation mode is usually a fundamental mode for such resonators [Shestopalov, 1991]. In addition to the fundamental modes, depending on the parameters of the open resonator and the diffraction grating, the other types of higher order oscillations (e.g., TEM02 ) occur influencing the coupling between the two open resonators through the diffraction fields.
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 f0 in the interval Af « ±1 GHz leads to the deviation of the main lobe direction from the normal, which increases the intensity of the diffraction field and, consequently, leads to the excitation of the second resonator at the edges of the frequency range. As the distance H increases, the coupling between the resonators becomes stronger reaching its maximum magnitude when the distances between the mirrors are equal to each other. In this case, oscillations in the second open resonator arise even at a frequency f ~ 46 GHz . 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 H ~(X/ 4 )(2 N +1), while the accumulation of energy inside the volume of the open resonator appears at values H ~(XN))2, where X is the radiated wavelength, N = 1, 2, … . The coupling in open resonators reaches its maximum when the distances between the resonators are approximately equal to each other, i.e., when the resonators are tuned to close frequencies. The typical response of the previously described coupled open resonators is presented in Fig. 4. Here P/Pmax is the power in the open resonators normalized to the maximum power Pmax . The resonance curve of a hemispherical open resonator is shown for comparison (curve 1) in the same figure. As can be seen from the presented data, the transmission band of coupled open resonators measured at the level of 0,5Pmax increases by a factor of nearly two, resulting in Af « 250 MHz. The resonance curves corresponding to coupled open resonators with reflective and strip diffraction gratings virtually coincide with each other under these conditions, which indicates the existence of efficient coupling in these systems through the diffraction of the fields at the periphery of the mirrors.
Fig. 4. Response of (1) a hemispherical open resonator and (2, 3) coupled open resonators with (2) metal-strip and (3) reflective diffraction gratings
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 TEMooq 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 f = 44,5 + 49,5 GHz for equal distances of spherical mirrors from the planes of the coupling element with a total distance between the spherical mirrors equal to H=31 mm. The achieved bandwidth measured at the 0,5Pmax power level is equal to Af «1,3 GHz . The narrowing of the transmission band of coupled open resonators observed in the higher frequency band (f=48,5 GHz) is due to the deviation of the radiation pattern for the diffraction grating-dielectric waveguide system from the normal and, consequently, the decrease in the coupling coefficient between the resonators.
Fig. 5. Response of (1) the spherical open resonator and (2) the system of resonators coupled through semitransparent diffraction gratings.
Analysis of the achieved bandwidth Af for the single resonator and coupled systems shows that the maximum bandwidth in systems with comparable H can be achieved when two open resonators are coupled through a strip diffraction grating. The bandwidth of the system with parallel open resonators is almost five times wider than the bandwidth of the system with series open resonators. It should be noted that the Q-factor of the coupled open resonators is of the same order as the Q-factor of the single open resonators. Therefore the open resonators coupled through the strip diffraction grating are preferable for systems requiring wideband operation. Such resonators also provide a reduced size of the system along the electron beam propagation axis.
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. 6a. It consists of a metal plane and a dielectric slab with a planar metallic diffraction grating on its surface.
£ is the permittivity of the dielectric. The source of electromagnetic energy is distributed along the grating. It excites various spatial harmonics of Cherenkov diffraction radiation of order n = 0, ± 1, ± 2,… and the power density Sn, which depends on the parameters of the structure. Fig. 6a demonstrates the excitation of Cherenkov ( S0£ ) and minus first diffraction ( S_1£ ) harmonics in the dielectric as well as minus first diffraction harmonic ( S_1o ) in open volume, which can be reflected back by a metal plane and fed to the metal-dielectric channel. A number of numerical and experimental methods for simulation of different excitation modes of Cherenkov diffraction radiation has been developed [Vorobyov et al., 1997, 2007]. They allow to determine the quantitative relation between the power densities of spatial harmonics in the structure as well as to optimize and tune their parameters.
Fig. 6. Quasi-optical resonators based on metal-dielectric slabs
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.
