Measured values of

## Abstract

Hemispherical open resonator (OR) with the segment of the oversized circular waveguide is considered. The cavity is formed by cylindrical, conical and spherical surfaces, and only axial-symmetric modes are excited. The power and spectral characteristics of such cavity with a dielectric bead and a rod have been studied. The quasi-periodic behavior of those dependencies was found out. Their qualitative agreement with similar dependencies for a cylindrical cavity is shown. It was found that physical processes in the cavity and in the hemispherical ОR with the segment of the oversized circular waveguide are identical. Dielectric permittivity and loss-angle tangent measurements have been carried out in millimeter wavelength range for the Teflon and Plexiglas samples having the shape of the bead as well as for the fused quartz and glass samples of the rod shape. It was found out that such a resonant system allows measuring samples with high losses, that is, especially important for quality control of food stuffs and analysis of biological liquids. Energy analysis of the ОR with the segment of the oversized rectangular waveguide has been performed. Basic possibility to apply such a resonant system for measurement of the dielectric permittivity of cylindrical samples with high losses has been shown as well.

### Keywords

- permittivity
- cavity
- open resonator
- circular waveguide
- rectangular waveguide
- oversized waveguide

## 1. Introduction

Measurement of electromagnetic properties of existing and novel materials in new frequency ranges is an important issue of the day. Recently emerged new class of artificial materials—composite materials (composites) are characterized by the negative refraction coefficient. Investigations showed that devices based on such materials can possess entirely unique properties and characteristics [1]. Caused by technology progress, advance of composite materials into millimeter and especially sub-millimeter ranges requires knowing of information about their electromagnetic characteristics. On the other hand, the real part of dielectric permittivity

Comparing the resonance curves corresponding to the cases of resonator with and without sample allows determining both

The studied sample having the shape of a bead is located in the bottom of the circular waveguide segment, in which there is a plane wave front of the propagating ТЕ_{01} mode. It allows measuring samples, the thickness of which exceeds the wavelength of the excited oscillation. At the research of substances with the application of the ОR having a cylindrical shape, difficulties related to their positioning in resonant volume may arise. At each measurement, the samples should be placed in the area with the same electric field intensity. The proposed resonator allows solving of this problem. The sample should be located along the ОR axis, where the electric field intensity of the excited oscillation is minimal. It provides analyzing of substances with high losses. In the case of the ОR having the segment of the oversized rectangular waveguide with the ТЕ_{10} mode, it is expedient to use the samples of a cylindrical shape. They should be located in the waveguide part parallel to the vector of the electric field intensity of the mode.

On the basis of the all above-stated, we can summarize that the goal of investigations, performed in this chapter, is theoretical and experimental research of the considered ОR, which will allow to measure, in millimeter and in sub-millimeter ranges, the electromagnetic characteristics of composite materials and biological liquids, as well as to control the quality of food stuff.

## 2. Open resonator with a dielectric bead

### 2.1. Resonator model

In the ОR, axial-symmetric modes are confined by caustics and hence they are with low diffraction losses. Placing of perfectly conducting boundary (Figure 1, dotted lines) in the area of exponentially vanishing intensity, almost does not affect the field pattern in OR. Our method is based on such physical principles.

Therefore, the task transforms to the study of the cavity resonator and approximate solution for the OR is achieved by selecting only modes with near axis distributed intensity (exponentially vanishing near conical boundary) from the cavity spectrum. We noticed that such approach for the electrodynamic model of the ОR was proposed in [4].

Let us consider the cavity as a body of revolution with perfectly conducting boundary and dielectric bead located in the bottom of the cylindrical part (Figure 1). We assume that the resonator is filled with a homogeneous isotropic medium having specific dielectric and magnetic conductivities

Here, the initial problem for Maxwell equations reduces to the problem of finding the wave numbers

which meet boundary conditions.

and conditions of the fields matching at the medium interface

Here,

Equations (1)-(4) with application of Bubnov-Galerkin’s method reduced to the system of the linear algebraic equations.

where

An approximate solution of the initial task [Equations (1)-(4)] could be represented as follows:

where

For measurement of the loss-angle tangent when a sample is placed in a cavity, both the resonator frequency shift and energy characteristic of the resonator are necessary to calculate [7]. For such calculations, it is required to find the following: the resonator

where

### 2.2. Numerical results

Dimensions of the considered cavity at numerical simulations have been chosen equal to sizes of the hemispherical ОR used in the experiment. The curvature radius of the spherical mirror is

The developed algorithm was validated using rigorous formulas for the spectrum of empty spherical volume resonator [9]. Evaluation of the algorithm convergence, related to the increase of the algebraic equation dimension, was carried out as well (Eq. (5)). As a result of such evaluation matrixes’ dimensions, _{0116} mode in the cavity with the bead made of Plexiglas having thickness _{0116} on the thickness of the dielectric bead will have a quasi-periodic behavior [11].

Dependencies of the frequency shift in the cavity on the thickness of the sample located at the bottom of cylindrical part are presented in Figure 2c. Curve 1 in Figure 2c corresponds to the bead made of Teflon (

Dependencies of the resonance frequency on the thickness of the dielectric beads made of Teflon (curve 1) and Plexiglas (curve 2) for cylindrical cavity are shown in Figure 2c by dotted curves. The diameter of this cavity is equal to the diameter of the cylindrical part of the studied cavity, and lengths of both the cavities coincide (Figure 2c). At the same time, a rigorous solution for a cylindrical cavity was obtained by application of the method based on the separation of variables. As it follows from Figure 2c, curves corresponding to both resonant systems qualitatively agree and have a quasi-periodic character. At the same time, a difference of eigen-frequencies of these cavities with beads having the same thickness and permittivity does not exceed 500 MHz.

Experimentally measured values of resonance frequency are shown in Figure 2c at the placement of beads made of Teflon (triangular marks) and Plexiglas (round marks) on the bottom of cylindrical part of the hemispherical ОR [12]. The difference of experimentally obtained values of resonance frequency from calculated by using developed electrodynamic model of the ОR does not exceed 50 MHz, and an error of the frequency measurement by using a resonant wavemeter in the considered frequency bandwidth is about 37 MHz [13]. With regard to the above, we can state validity of the proposed electrodynamic model of the cavity (Figure 1) to measure the electromagnetic parameters of substances using the ОR [3].

As shown earlier (Figure 2c), dependencies of the resonance frequency on the bead thickness at the constant value of its permittivity have a quasi-periodic character. Dependencies of the resonance frequency on permittivity of the bead having constant thickness should look similar. It can be explained by the fact that with a change of

Dependencies of TE_{0116} mode resonant frequency

As an important characteristic, needed for valuation of the loss-angle tangent for dielectric samples, is

From the presented diagram one can see that behavior of the

In the investigation of the samples with high losses, a

Here, it should be noted that in the

Thus, analysis of the energy and spectral characteristics of the cavity with dielectric inclusions, formed by the cylindrical, conical and spherical surfaces was carried out in this subsection. As a result of the performed study, it was found out that physical processes in the considered cavity and in the hemispherical ОR with the segment of the oversized circular waveguide having dielectric beads are identical. It allows to conclude that the proposed model is valid for the resonator to measure electromagnetic parameters of substances in the millimeter range of the wavelengths.

### 2.3. Measurement of the permittivity and losses in the samples

The hemispherical ОR, formed by the spherical 13 and flat 14 mirrors having diameter 38 mm (Figure 4) [6, 14], is used for measurements. Short-segment of the oversized circular waveguide 15 of diameter 18 mm is located in the center of the flat mirror. The studied sample 17 having the shape of a bead is placed at the waveguide plunge 16. Distance from the flat mirror to the plunge is equal to _{10} mode electric field in the rectangular waveguides is orthogonal to the plane of Figure 4.

The distance _{01q} mode in the plane of the spherical mirror and is equal to 5.5 mm (Figure 4). In this case ТЕ_{01q} mode is excited with maximal efficiency. For the isolation of the generator G4–142 and the resonator, additional setting attenuator 2 is included in the circuit. The tuning to the resonance is implemented by moving the spherical mirror 13 along the resonator axis. The distance between the reflectors is evaluated by using a measuring projective device having accuracy of ±0.001 mm. The signal extraction from the ОR is performed by using the second slot coupling element, which, as stated earlier, is on the spherical mirror and has the same dimensions as the first one, and is located at the distance 11 mm from it.

In the circuit, an additional receiving transmission line is included for the measurement of the reflection coefficient from the resonator. This transmission line comprises a directional coupler 3, a measuring polarizing attenuator 4, a crystal detector 6, a resonant amplifier 7, tuned to the frequency of modulating voltage and an oscillograph 8. Reflectivity is measured in the plane, in which input impedance of the resonator with a certain part of the waveguide is purely active [15].

The reflectivity factor on voltage is defined by the formula

The validation of the proposed method of permittivity measurement using the considered ОR (Figure 4) was performed with the samples having the shape of dielectric beads made of Teflon and Plexiglas. The diameter of the beads was equal to

We measured _{0110} is excited. At the same time, the distance between the mirrors is equal to 22.139 mm (_{0110} mode of the hemispherical ОR is being transformed into the axial-symmetric mode ТЕ_{0116} [3]. In that case, the resonant distance is being measured already between the spherical ОR mirror and the plunge.

Now the bead 17 having the thickness 2.99 mm and the diameter 18 mm, made of Teflon is placed on the plunge 16. Frequency of the generator G4–142 is tuned to achieve the resonant response. In the resonator at the frequency _{0116} mode is excited, that is identified by using the perturbation technique. Getting the value of the resonance frequency, we can evaluate permittivity of the studied sample with thickness 2.99 mm. For that purpose, we use the curve 1 (Figure 3a). In a similar way, we evaluate permittivity of the sample made of Teflon and having thickness 3.58 mm. In this situation, placing the sample on the plunge located in the cylindrical part of the ОR, we got the value of the resonance frequency 68.410 GHz for the ТЕ_{0116} mode. In order to value _{0116} is excited but now only at the frequencies 68.051 GHz and (

Material | Thickness of the sample | The measured value | Literary value | The difference | |
---|---|---|---|---|---|

Teflon | 2.99 | 2.085 ± 0.020 | 2.07 ± 0.04 [10] | 0.7% | 2.705 |

Teflon | 3.58 | 2.124 ± 0.020 | 2.07 ± 0.04 [10] | 2.5% | 2.962 |

Plexiglas | 2.99 | 2.599 ± 0.026 | 2.557 ± 0.026 [10] | 1.6% | 3.321 |

Plexiglas | 3.58 | 2.616 ± 0.026 | 2.557 ± 0.026 [10] | 2.2% | 4.342 |

In Table 1, _{0116} mode when putting the sample into ″empty″ ОR.

In the next step, the dielectric losses in the samples made of Teflon and Plexiglas and having thickness 2.99 mm are evaluated. For their finding, one should calculate the energy filling factor of the resonator with the sample on the electric field

The behavior

In calculations, we assume that there is nondegenerate mode TE_{0116} exists in the considered resonator. In Figure 6, also by dotted shows the factor _{0116} is excited. The diameter of the resonator is equal to the diameter of the cylindrical part of the considered ОR model, and lengths of the both resonators coincide (Figure 2c). The studied samples of various thicknesses are placed on the end cover of the cylindrical cavity. From the presented dependencies one can see that for the samples of various thickness in whole range of the

Now let us evaluate losses in the bead having diameter 18 mm and

where

In the absence of the measured sample in the resonant volume, unloaded

where

If we deduct (Eq. (11)) from (Eq. (10)), then, we will get the relation determining the dielectric losses tangent in the sample.

Diffraction

For the samples under studies having the shape of the bead (

In order to find _{0116} mode excitation. At the same time, loaded and unloaded

where _{0116} mode in the considered resonator.

Based on the research carried out with a metal screen covering the ОR [18], it was determined that excitation efficiency of the considered ТЕ_{0116} mode can be accepted to be equal to 0.96. Since two slot coupling elements having identical dimensions (3.6 × 0.14 mm) and located symmetrically to the resonator axis are used for excitation of the resonator and the signal output into the load, the input and output coupling should be the same, that is, * Г* performed using the directional coupler (

For calculation of the loss-angle tangent in the samples made of Teflon and Plexiglas, we calculate the ohmic _{0116} exists, turned out to be equal to 1660. At the same time, the coupling coefficient _{00} = 2801.

Taking into account the obtained values

Material | ||||||
---|---|---|---|---|---|---|

Teflon | 0.0886 | 49,339 | 1588 | 0.287 | 2604 | (3.117 ± 0.034) × 10^{−4} |

Plexiglas | 0.0878 | 48,005 | 502 | 0.181 | 712 | (1.192 ± 0.035) × 10^{−2} |

As may be seen in Table 2, measured values of ^{−4} [20], which differs by 11% from the result obtained using the proposed resonator cell. In the case of the material with high losses (Plexiglas), obtained value of ^{−2}) [20].

The authors did not aim to get high accuracy

## 3. Open resonator with a dielectric rod

### 3.1. Resonator model

The method of the solution is based on the same physical principles as in the case of the resonator with the dielectric bead. Thus, we consider a resonance cavity with the boundary made of a spherical, a conical and a cylindrical perfectly conducting surface. There is a cylindrical rod extended all along the resonator axis (area 2 in Figure 7), which is assumed to be a homogeneous and isotropic medium, having the material parameters

We confine ourselves to the analysis of axially-symmetric oscillations of TE-mode, which in cylindrical coordinates, where axis

that satisfy the boundary conditions:

and the field matching conditions at

Here,

The numerical algorithm for the problem (Eq. (16))÷(Eq. (19)) utilizes the method of Bubnov-Galerkin, as described earlier, and the problem is reduced to the system of linear algebraic equalizations (Eq. (5)) [6, 21].

The developed algorithm, as in the case with the dielectric bead, was tested by the passing to the limit from the geometry of the considered resonator to the cavities of spherical and cylindrical shape. The correctness of the described approach is validated by the papers [9, 12, 22]. Moreover, the algorithm convergence rate was estimated numerically for the growing dimensional representation of the algebraic problem (Eq. (5)).

### 3.2. Numerical and experimental results

The computations have been carried out for a resonator having the same dimensions, as in the case of the resonator with the bead. In Figure 8, the lines of equal amplitudes _{0115} in the resonator with rods, made of Teflon and having permittivity _{0115} mode resonance frequencies at the increase of the rod diameter from 1 to 2 mm is insignificant. It decreases from _{0115} in the resonator with the same dimensions for the rod, made of fused quartz (

The carried out research (Figure 8c) demonstrated the strong relation of the sample with the electromagnetic field and high filling coefficient of the resonator on the electromagnetic field [7]. In such a manner, for qualitative control of liquid samples, the composition of which includes water, it is necessary to use pipes, made of the material having low permittivity, as compared with the substance under studies.

On the basis of the analysis carried out, one can say that sensitivity of the resonant cell is defined by the diameter of the cylindrical sample and by the value of its permittivity. From Figure 8, one can see that electric field intensity near the conical metal surface (dotted lines) is low. It indicates that the considered cavity is equivalent to the ОR, in which high-

The experimental measurements of the permittivity of materials can be performed with the aid of calibration curves, that is, the dependencies of the resonator frequency shift on the permittivity of the cylindrical samples of various diameters, introduced into the resonance cavity. These characteristics are shown in Figure 9.

The upper part of the figure presents the series of curves plotted for the ТЕ_{0116} mode by the rod diameter of 2 mm (curve 1) and 1.5 mm (curve 2). The bottom part presents the ТЕ_{0115} mode by the same diameters of the samples, that is, 2 mm (curve 3) and 1.5 mm (curve 4). The dotted lines in the same figure, being almost parallel to those described earlier, illustrate similar dependencies for ТЕ_{0116} and ТЕ_{0115} modes in a cylindrical resonance cavity, with the cylindrical test pieces of the said size arranged along the resonator axis.

The length and the diameter of the cylindrical resonator were chosen to be equal to the length of the considered resonator and to the diameter of its cylindrical part (Figure 7). The problem considering the cylindrical resonator with a rod was solved using the method of variable separation, that is, its resonance frequency was described by rigorous formulas.

The figure demonstrates an obvious similarity of the curves for the both resonator types. It proves the fact that the nature of the physical processes occurring in the resonators is similar, both in the resonator under consideration and in the cylindrical cavity. It is an indirect evidence of the adequacy of our theoretical considerations. One more point to emphasize is that small changes of samples’ permittivity, which are critical for oil quality control and food stuff, should be measured at the areas with a stronger disperse dependence, which is solely up to the diameter of the cylindrical sample (Figure 9).

The block diagram of the experimental unit used in the research is given in Figure 4. Only, in this case, the cylindrical sample was used. A sample was inserted into the cavity through a hole in the middle of the waveguide plunger, with a guide for the precise alignment of the sample along the resonator axis.

The measured shift of the resonance frequencies and the calibration curves in Figure 9 were used to determine the dielectric permittivities of two cylindrical samples made of fused quartz and silicate glass. The value of resonance frequencies obtained in the experimental measurements for ТЕ_{0116}- and ТЕ_{0115}-modes for the case when the cylindrical samples of 2 mm and 1.5 mm in diameter were located along the resonator axis are marked with squares at the calculated curve. The results of measuring the dielectric permittivity of cylindrical samples of various diameters are listed in Table 3.

## 4. Open resonator with a segment of rectangular waveguide

A new electrodynamic system appears when inserting the segment of the short-circuited rectangular waveguide in the center of one of the ОR mirrors [24]. Cross-section sizes of the waveguide _{10} mode excitation by the fundamental mode ТЕМ_{00q}. One can consider such ОR as a resonant cell for measurement of composite materials and biological liquids electromagnetic specifications as well as to control the quality of food stuff in millimeter and in sub-millimeter ranges.

We consider the hemispherical ОR with a rectangular waveguide located in the center of the flat mirror. Reflection from the waveguide horn is neglected. We consider the resonator mirrors apertures as an infinite one. Omitting intermediate computations, we write down in the final form the expression, determining efficiency of the TE_{10} mode excitation in the waveguide of the OR, using the TEM_{00q} mode [25].

Here,

Dependence _{10} mode excitation in the rectangular waveguide, located in the center of the ОR flat mirror, using ТЕМ_{00q} fundamental mode of the resonator is maximal and equal to 0.881.

As a result of the theoretical analysis, it was demonstrated that the efficiency of the ТЕ_{10} mode excitation in the segment of the rectangular waveguide, using the ОR fundamental oscillation, can amount the value about 90%. Therefore, such resonant system should have good selective properties, that is, an advantage for the analysis of dielectric samples with high losses. Besides, since the cross-section sizes

In such ОR, losses should increase, since ohmic losses in the walls of the rectangular waveguide segment are added. It would result in a decrease of the loaded

Block diagram of the experimental setup, which was used for carrying out research of the hemispherical ОR with the segment of the oversized rectangular waveguide, is shown in Figure 11.

The ОR is formed by the flat mirror 5, having aperture 60 mm, and by the spherical focusing mirror 4, having a curvature radius _{10} mode excitation by the fundamental mode of the ОR. As it turned out, for presented dimensions of the resonator at

The resonator is excited by the slot coupling element, having sizes 3.6 × 0.16 mm, located in the center of the spherical mirror. The adjusting attenuator 2 is included into the setup for decoupling of the frequency generator and the resonator. Alignment to resonance is implemented by moving the spherical mirror 4 with the elements of the waveguide along the resonator axis. The input waveguide is oriented in such a manner that the vector ** E** of the fundamental mode ТЕ

_{10}is orthogonal to the plane of the drawing (Figure 11).

Receiving transmission line consists of the auxiliary line of the directional coupler 3, measuring polarizing attenuator 8, detector 10, resonant amplifier 11 and oscillograph 12. The resonant wavemeter 9 is included into the setup for monitoring frequency of the high-frequency generator 1. A double-stub matcher 13 is installed in the branch of the matched load 14 of the directional couplers 3. The photo of the OR and the experimental unit is represented in Figure 12.

The above-described procedure is used for computing of the resonant reflection factor of the resonator. Results of measurement of the resonant reflection factor on the distance between mirrors _{00q} mode is excited. Identification of the oscillations modes was performed using the perturbation technique [13]. The technique described in [27] was applied for the definition of the

As can be seen from Figure 13, with decrease of the distance between mirrors, the reflection factor from the resonator diminishes. It is related to the reduction of the diffraction and ohmic losses in the resonant system. Exceptions include the cases of the interaction of the considered oscillation with other oscillations excited in the ОR (_{00q} mode at

Dependence _{10} excitation by means of the ОR mode ТЕМ_{0015}. The difference between experimentally obtained value

The dependence of reflection from resonator on the length of the oversized rectangular waveguide segment for certain mode is of the practical interest as well. We assume that in the hemispherical ОR (Figure 13, curve 1), a no degenerate mode should exist. Moreover, the distance between resonator mirrors should correspond to low diffraction losses. Therefore, in terms of the diagram presented in Figure 13, we choose the mode ТЕМ_{0016} (

From the figure, one can see that moving the plunge from the surface of the flat mirror (

Considered here resonant system will be the most promising at the analysis of biological liquids and food stuffs, the basic element of which is water (wines, juices, drinks). For carrying out measurements, a pipe made of the material of lower permittivity than that of a sample is placed into the oversized rectangular waveguide parallel to the vector of electric field intensity of the ТЕ_{10} mode. For reduction of losses inserted to the resonant system, it can be displaced to one of the side walls of the waveguide.

## 5. Conclusions

The cavity with the dielectric layer, being an electrodynamic model of the hemispherical ОR with the segment of the oversized circular waveguide and dielectric bead, is considered in this chapter. As a result of the carried out theoretical analysis, it is shown that dependence of the frequency upon the thickness of the dielectric bead, located on the bottom of the cavity cylindrical part, has a quasi-periodic behavior. Such behavior is related to the amplitude distribution of

The hemispherical ОR, with the segment of the rectangular oversized waveguide located in the center of the flat mirror, has been considered in this chapter as well. As a result of the carried out theoretical analysis, it was demonstrated that efficiency of the ТЕ_{10} mode excitation in such waveguide by means of the resonator fundamental mode ТЕМ_{00q} can reach the value ~90% at cross-section sizes _{10} mode. For reduction of the losses inserted into the resonant system, it can be displaced to one of the side walls of the waveguide.

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