2. Open resonator with a dielectric bead

#### 2.1. Resonator model

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 ε<sup>0</sup> and dielectric loss-angle tangent tan δ of water, which is the main component of the whole series of food stuff and biological liquids, decrease with the wavelength shortening. Therefore, for effective control of their quality, we also should move to the specified ranges of the wavelengths. For measurement of electromagnetic characteristics of such substances, the application of the resonant techniques is needed due to their higher accuracy. The main point of such methods consists of observation of resonant curves of the oscillatory circuit, in which

28 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

Comparing the resonance curves corresponding to the cases of resonator with and without sample allows determining both ε<sup>0</sup> and tg δ using the Q-factor and frequency shift. Open resonators (ОR) are used as circuits in millimeter and sub-millimeter ranges for such measurements. The peculiarity of such resonant systems consists of the fact that, apart from high Qfactor, their geometrical dimensions account a few tens of wavelengths, and coupling with free space provides an additional mode selection and free access to the resonant volume. However, such resonant systems are applicable to use just for investigations of substances with low losses. In the case of high losses, sample thickness should not exceed the size of the skin layer since it can result in oscillation suppression in such resonant systems. This circumstance imposes limitations to the application of the ОR in the research of electromagnetic characteristics of composite materials and biological liquids, quality control of food stuffs since they are characterized by high losses. Therefore, the most promising resonant system to use for investigations of such substances is the ОR proposed in [2, 3]. It represents symbiosis of the ОR and the segment of the oversized waveguide, which could be both circular and rectangular. A distinctive feature of such resonant systems is that they are characterized by the single frequency response in the wide range of frequencies [3]. It is an advantage at investigations of electromagnetic characteristics of substances. At placement of the sample in the waveguide part of the ОR, the measurement accuracy increases due to keeping high Q-factor, and there-

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<sup>01</sup> 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<sup>10</sup> 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

the sample of the studied substance is placed.

fore, the range of the analyzed values ε<sup>0</sup> and tanδ extends.

electric field intensity of the mode.

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 ε1, μ<sup>1</sup> in the area (1) and ε2, μ<sup>2</sup> in the area (2). We consider only axialsymmetric TE modes with Eφ, Hr and Ez are components of the electromagnetic field in the cylindrical coordinate system with the axis z, coinciding with the axis of symmetry.

Figure 1. Geometric model of the OR with the dielectric bead.

Here, the initial problem for Maxwell equations reduces to the problem of finding the wave numbers k ¼ ω=c (where с is the speed of the light in vacuum and ω is a circular frequency), for which exist non-zero solutions U<sup>1</sup> and U<sup>2</sup> of the two-dimensional Helmholtz's equations.

$$\Delta\_{rz}l I\_1 + \left(k^2 \varepsilon\_1 \mu\_1 - \left(1/r^2\right)\right) l I\_1 = 0, \ h\_d < z < l, \ 0 < r < b(z), \tag{1}$$

$$
\Delta\_{rz} \mathcal{U}\_2 + \left(k^2 \varepsilon\_2 \mu\_2 - \left(1/r^2\right)\right) \mathcal{U}\_2 = 0, \ 0 < z < h\_d, \ 0 < r < b(z), \tag{2}
$$

which meet boundary conditions.

$$\begin{cases} \mathcal{U}\_1(r,l) = 0, & \mathcal{U}\_2(r,0) = 0, \\ \mathcal{U}\_{1,2}(b(z),z) = 0, & \mathcal{U}\_{1,2}(0,z) = 0, \end{cases} \tag{3}$$

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 Q-factor Q<sup>ε</sup> caused by losses in the sample; energy factor KE of the resonator filling by electric field [7] and an

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

Q<sup>ε</sup> ¼ �2ω<sup>0</sup>

KEð Þ¼� <sup>ε</sup> <sup>2</sup> <sup>ε</sup><sup>0</sup>

ð

0 @

V j j <sup>H</sup> <sup>2</sup>

where Δ<sup>R</sup> is depth of the electromagnetic field penetration into metal; V,S are volume and area of the resonator surface; H is the distribution of the magnetic components of the electromagnetic field in the volume of the resonator; H<sup>τ</sup> is the tangent component of the magnetic. Distributions of magnetic components H and H<sup>τ</sup> of the electromagnetic field are calculated

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 R ¼ 39 mm, aperture of this mirror is 2a<sup>1</sup> ¼ 38 mm, the diameter of the circular waveguide segment and dielectric samples is 2a<sup>2</sup> ¼ 18 mm, the length of the resonator cylindrical part 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, A and B from (Eq. (5)) have been chosen to be equal to N ¼ P � I at P ¼ 60, I ¼ 6 (Eq. (6)). Further increase of P and I does not provide considerable changes at calculation results of the eigenvalues and the eigenvectors of the task (Eq. (5)). The examples of distribution for an electric field component of the TE0116 mode in the cavity with

presented in Figure 2a [10]. At the same time, frequency of resonant oscillation decreases down to the value f ¼ 68.052 GHz as compared to f ¼ 71.382 GHz for an empty resonator with the same dimensions. For the bead made of Plexiglas having the same diameter, but thickness hd ¼ 3.6 mm, resonance frequency decreases down to the value f ¼ 66.999 GHz (Figure 2b). As can be seen in Figure 2a and b, depending on the dielectric bead thickness, its top boundary can coincide both with the node and the antinode of the electric field component of the standing wave in the cavity. Therefore, it should be expected that in such a resonant

the bead made of Plexiglas having thickness hd ¼ 2.75 mm and permittivity ε<sup>0</sup>

2 ω0 ∂ω<sup>0</sup> ∂ε<sup>0</sup> 2

dV�ð

S

j j <sup>H</sup><sup>τ</sup> <sup>2</sup> dS

1

=ω00, (7)

, (8)

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31

A, (9)

<sup>2</sup> ¼ 2.61 are

ohmic Q-factor QR<sup>ε</sup> of the resonator with the sample [8]. At the same time,

QR<sup>ε</sup> ¼ ð Þ 2=Δ<sup>R</sup>

with the application of the developed theoretical model.

h ¼ 12.434 mm, the resonator length is l ¼ 35.295 mm (Figure 1).

2.2. Numerical results

and conditions of the fields matching at the medium interface z ¼ hd.

$$\mathcal{U}l\_1(r,h\_d) = \mathcal{U}l\_2(r,h\_d), \quad \frac{1}{\mu\_1} \frac{\partial \mathcal{U}l\_1(r,h\_d)}{\partial \mathbf{z}} = \frac{1}{\mu\_2} \frac{\partial \mathcal{U}l\_2(r,h\_d)}{\partial \mathbf{z}}.\tag{4}$$

Here, U1, <sup>2</sup>ð Þ r; z is azimuthal component of electric field of a mode; function b zð Þ is boundary surface, which is supposed to be piecewise-differentiable in the interval 0 < z < l, where l is the length of the cavity; hd is the thickness of the dielectric bead; <sup>Δ</sup>rz <sup>¼</sup> <sup>∂</sup><sup>2</sup> <sup>=</sup>∂r<sup>2</sup> <sup>þ</sup> <sup>∂</sup>=r∂<sup>r</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> =∂z<sup>2</sup> is the two-dimensional Laplace's operator; ð Þ r;φ; z are cylindrical coordinate system with axis z, coinciding with the axis of the cavity symmetry.

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

$$\mathbf{AC} + k^2 \mathbf{BC} = 0,\tag{5}$$

where <sup>C</sup> <sup>¼</sup> ð Þ cn <sup>N</sup> <sup>n</sup>¼<sup>1</sup> is the vector-column of unknown factors, <sup>A</sup> <sup>¼</sup> k k amn <sup>N</sup> m,n¼<sup>1</sup> and <sup>B</sup> <sup>¼</sup> k k bmn <sup>N</sup> m,n¼1 are matrixes with matrix elements, which are prescribed by the formulas presented in [5, 6].

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

$$E(r,z) = \begin{cases} \sum\_{i=1}^{l} \lambda\_1 \binom{\mu\_i r}{b} \sum\_{p=1}^{p} c\_p^i \frac{\mu\_1 \varepsilon\_2}{\mu\_2 \varepsilon\_1} \frac{\cos \lambda\_p \sqrt{\varepsilon\_1 \mu\_1} h\_d}{\cos \lambda\_p \sqrt{\varepsilon\_1 \mu\_1} (l-h\_d)} \sin \lambda\_p \sqrt{\varepsilon\_1 \mu\_1} (l-z), & h\_d < z < l, \\\sum\_{i=1}^{l} \lambda\_1 \binom{\mu\_i r}{b} \sum\_{p=1}^{p} c\_p^i \sin \lambda\_p \sqrt{\varepsilon\_1 \mu\_1} z, & 0 < z < h\_d, \\\ \omega = kc, \end{cases} \tag{6}$$

where <sup>k</sup> and <sup>C</sup><sup>N</sup> are solutions of the [Eq. (5)]; <sup>ω</sup> <sup>¼</sup> <sup>ω</sup><sup>0</sup> <sup>þ</sup> <sup>i</sup>ω<sup>00</sup> is complex frequency of the natural oscillation and N ¼ P � I is the dimension of the algebraic task.

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 Q-factor Q<sup>ε</sup> caused by losses in the sample; energy factor KE of the resonator filling by electric field [7] and an ohmic Q-factor QR<sup>ε</sup> of the resonator with the sample [8]. At the same time,

$$Q\_{\varepsilon} = -2\omega'/\omega'',\tag{7}$$

$$K\_E(\varepsilon) = -2 \frac{\varepsilon\_2'}{\omega'} \frac{\partial \omega'}{\partial \varepsilon\_2'},\tag{8}$$

$$Q\_{R\varepsilon} = (2/\Delta\_R) \left( \int\_V |\mathbf{H}|^2 dV \bigg/ \int\_S |\mathbf{H}\_{\mathbf{r}}|^2 dS \right) . \tag{9}$$

where Δ<sup>R</sup> is depth of the electromagnetic field penetration into metal; V,S are volume and area of the resonator surface; H is the distribution of the magnetic components of the electromagnetic field in the volume of the resonator; H<sup>τ</sup> is the tangent component of the magnetic. Distributions of magnetic components H and H<sup>τ</sup> of the electromagnetic field are calculated with the application of the developed theoretical model.

#### 2.2. Numerical results

Here, the initial problem for Maxwell equations reduces to the problem of finding the wave numbers k ¼ ω=c (where с is the speed of the light in vacuum and ω is a circular frequency), for which exist non-zero solutions U<sup>1</sup> and U<sup>2</sup> of the two-dimensional Helm-

U1ð Þ¼ r; l 0, U2ð Þ¼ r; 0 0,

U1, <sup>2</sup>ð Þ¼ b zð Þ; z 0, U1,2ð Þ¼ 0; z 0,

∂U1ð Þ r; hd <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>1</sup>

μ1

Here, U1, <sup>2</sup>ð Þ r; z is azimuthal component of electric field of a mode; function b zð Þ is boundary surface, which is supposed to be piecewise-differentiable in the interval 0 < z < l, where l is

is the two-dimensional Laplace's operator; ð Þ r;φ; z are cylindrical coordinate system with axis

Equations (1)-(4) with application of Bubnov-Galerkin's method reduced to the system of the

AC þ k 2

<sup>n</sup>¼<sup>1</sup> is the vector-column of unknown factors, <sup>A</sup> <sup>¼</sup> k k amn <sup>N</sup>

cosλ<sup>p</sup>

cosλ<sup>p</sup>

ffiffiffiffiffiffiffiffiffi ε1μ<sup>1</sup>

are matrixes with matrix elements, which are prescribed by the formulas presented in [5, 6].

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

ffiffiffiffiffiffiffiffiffi ε1μ<sup>1</sup> <sup>p</sup> ð Þ <sup>l</sup> � hd

p z, 0 < z < hd,

where <sup>k</sup> and <sup>C</sup><sup>N</sup> are solutions of the [Eq. (5)]; <sup>ω</sup> <sup>¼</sup> <sup>ω</sup><sup>0</sup> <sup>þ</sup> <sup>i</sup>ω<sup>00</sup> is complex frequency of the natural

ffiffiffiffiffiffiffiffiffi ε1μ<sup>1</sup> p hd

sinλ<sup>p</sup>

ffiffiffiffiffiffiffiffiffi ε1μ<sup>1</sup>

<sup>2</sup> � � � � <sup>U</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>, hd <sup>&</sup>lt; <sup>z</sup> <sup>&</sup>lt; l, <sup>0</sup> <sup>&</sup>lt; <sup>r</sup> <sup>&</sup>lt; b zð Þ, (1)

<sup>2</sup> � � � � <sup>U</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>0</sup> <sup>&</sup>lt; <sup>z</sup> <sup>&</sup>lt; hd, <sup>0</sup> <sup>&</sup>lt; <sup>r</sup> <sup>&</sup>lt; b zð Þ, (2)

μ2

∂U2ð Þ r; hd

BC ¼ 0, (5)

<sup>∂</sup><sup>z</sup> : (4)

<sup>=</sup>∂r<sup>2</sup> <sup>þ</sup> <sup>∂</sup>=r∂<sup>r</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

m,n¼<sup>1</sup> and <sup>B</sup> <sup>¼</sup> k k bmn <sup>N</sup>

<sup>p</sup> ð Þ <sup>l</sup> � <sup>z</sup> , hd <sup>&</sup>lt; <sup>z</sup> <sup>&</sup>lt; l,

(3)

=∂z<sup>2</sup>

m,n¼1

(6)

holtz's equations.

<sup>Δ</sup>rzU<sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

ΔrzU<sup>2</sup> þ k

which meet boundary conditions.

2

�

z, coinciding with the axis of the cavity symmetry.

linear algebraic equations.

where <sup>C</sup> <sup>¼</sup> ð Þ cn <sup>N</sup>

E rð Þ¼ ; z

X I

8 >>>>>><

>>>>>>:

i¼1 J1 μi r b � �X P

X I

i¼1 J1 μi r b � �X P

p¼1 ci p μ1ε<sup>2</sup> μ2ε<sup>1</sup>

p¼1 ci <sup>p</sup>sinλ<sup>p</sup>

oscillation and N ¼ P � I is the dimension of the algebraic task.

ω ¼ kc,

ε1μ<sup>1</sup> � 1=r

30 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

ε2μ<sup>2</sup> � 1=r

and conditions of the fields matching at the medium interface z ¼ hd.

<sup>U</sup>1ð Þ¼ <sup>r</sup>; hd <sup>U</sup>2ð Þ <sup>r</sup>; hd , <sup>1</sup>

the length of the cavity; hd is the thickness of the dielectric bead; <sup>Δ</sup>rz <sup>¼</sup> <sup>∂</sup><sup>2</sup>

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 R ¼ 39 mm, aperture of this mirror is 2a<sup>1</sup> ¼ 38 mm, the diameter of the circular waveguide segment and dielectric samples is 2a<sup>2</sup> ¼ 18 mm, the length of the resonator cylindrical part is h ¼ 12.434 mm, the resonator length is l ¼ 35.295 mm (Figure 1).

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, A and B from (Eq. (5)) have been chosen to be equal to N ¼ P � I at P ¼ 60, I ¼ 6 (Eq. (6)). Further increase of P and I does not provide considerable changes at calculation results of the eigenvalues and the eigenvectors of the task (Eq. (5)). The examples of distribution for an electric field component of the TE0116 mode in the cavity with the bead made of Plexiglas having thickness hd ¼ 2.75 mm and permittivity ε<sup>0</sup> <sup>2</sup> ¼ 2.61 are presented in Figure 2a [10]. At the same time, frequency of resonant oscillation decreases down to the value f ¼ 68.052 GHz as compared to f ¼ 71.382 GHz for an empty resonator with the same dimensions. For the bead made of Plexiglas having the same diameter, but thickness hd ¼ 3.6 mm, resonance frequency decreases down to the value f ¼ 66.999 GHz (Figure 2b). As can be seen in Figure 2a and b, depending on the dielectric bead thickness, its top boundary can coincide both with the node and the antinode of the electric field component of the standing wave in the cavity. Therefore, it should be expected that in such a resonant system, dependence of the oscillation frequency TE0116 on the thickness of the dielectric bead will have a quasi-periodic behavior [11].

point (b) in Figure 2c), the dispersive dependence of the resonator frequency on the sample

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

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

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

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.

component of the standing wave appear periodically on the top of the bead, and condition of equality of the bead thickness to integer number of half wavelength does not hold for any ε<sup>0</sup>

Dependencies of TE0116 mode resonant frequency f of the considered cavity (Figure 1) on the permittivity of the beads having various thicknesses hd are presented in Figure 3a. Curve 1 is for the bead, with thickness of 2.99 mm, and curve 2 is for the bead having thickness 3.58 mm.

Figure 3. Dependencies of the resonant frequency on the permittivity of the beads having a various thickness (a) and of

<sup>2</sup> the nodes and antinodes of electric

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33

2.

measure the electromagnetic parameters of substances using the ОR [3].

It can be explained by the fact that with a change of ε<sup>0</sup>

the Q-factor of the cavity on the thickness of the sample (b).

thickness has a large steepness.

does not exceed 500 MHz.

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 (ε<sup>0</sup> <sup>2</sup> ¼ 2.07 [10]), and curve 2 is for the bead made of Plexiglas. Marks (а) and (b) at the curve 2 show the beads' thicknesses, corresponding to the field distributions presented in Figure 2a and b. Apparently, the weak dependence of the frequency and the bead thickness takes place in the case, when the node of the electric field component of the standing wave in the cavity is located near to the top of the sample (Figure 2a, point (а) in Figure 2c). In the case of electric components antinode location near to the top of the sample (Figure 2b,

Figure 2. The examples of distribution of the electric field component of the ТЕ<sup>0116</sup> mode in the cavity with beads from Plexiglas (а, b) and dependencies of the resonance frequency on the thickness of beads made of different material for two types of cavities (c).

point (b) in Figure 2c), the dispersive dependence of the resonator frequency on the sample thickness has a large steepness.

system, dependence of the oscillation frequency TE0116 on the thickness of the dielectric bead

32 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

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

and (b) at the curve 2 show the beads' thicknesses, corresponding to the field distributions presented in Figure 2a and b. Apparently, the weak dependence of the frequency and the bead thickness takes place in the case, when the node of the electric field component of the standing wave in the cavity is located near to the top of the sample (Figure 2a, point (а) in Figure 2c). In the case of electric components antinode location near to the top of the sample (Figure 2b,

Figure 2. The examples of distribution of the electric field component of the ТЕ<sup>0116</sup> mode in the cavity with beads from Plexiglas (а, b) and dependencies of the resonance frequency on the thickness of beads made of different material for two

<sup>2</sup> ¼ 2.07 [10]), and curve 2 is for the bead made of Plexiglas. Marks (а)

will have a quasi-periodic behavior [11].

bead made of Teflon (ε<sup>0</sup>

types of cavities (c).

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 ε<sup>0</sup> <sup>2</sup> the nodes and antinodes of electric component of the standing wave appear periodically on the top of the bead, and condition of equality of the bead thickness to integer number of half wavelength does not hold for any ε<sup>0</sup> 2.

Dependencies of TE0116 mode resonant frequency f of the considered cavity (Figure 1) on the permittivity of the beads having various thicknesses hd are presented in Figure 3a. Curve 1 is for the bead, with thickness of 2.99 mm, and curve 2 is for the bead having thickness 3.58 mm.

Figure 3. Dependencies of the resonant frequency on the permittivity of the beads having a various thickness (a) and of the Q-factor of the cavity on the thickness of the sample (b).

Dependencies in Figure 3a corresponding to the sample thickness provide defining a real part of the samples permittivity ε<sup>0</sup> <sup>2</sup> by resonant frequency shift.

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

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

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35

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 3λ<sup>w</sup> (λw-waveguide wavelength). On the surface of the spherical mirror of curvature radius R ¼ 39 mm, two coupling slots are located, through one of which the signal with 1 kHz amplitude modulation from the high-frequency generator G4–142 inputs into the resonant volume, and through the second one, signal outputs to the load. The slots are tapering from the main section of the rectangular waveguide 3.6 � 1.8 mm into the narrow one 3.6 � 0.14 mm. Both coupling elements are oriented in such a manner that the vector of TE10 mode electric field in the rectangular waveguides is orthogonal to the plane of Figure 4.

2.3. Measurement of the permittivity and losses in the samples

of the wavelengths.

Figure 4. Block diagram of the experimental setup.

As an important characteristic, needed for valuation of the loss-angle tangent for dielectric samples, is Q-factor Qε, caused by losses in a dielectric (Eq. (7)), and an ohmic Q-factor QR<sup>ε</sup> of the resonator with (Eq. (9)). Dependencies of Q<sup>ε</sup> (solid curves) and QR<sup>ε</sup> (dotted curves) on the beads' thickness hd calculated using the developed theoretical model are presented in Figure 3b. The Q-factor Q<sup>ε</sup> of the cavity with the sample of the shape of a bead, permittivity of which ε<sup>0</sup> <sup>2</sup> <sup>¼</sup> 2.07 and loss-angle tangent tan<sup>δ</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> , is designated by the curve 1. The curves 2 and 3 correspond to the ohmic Q-factor of the resonator QR<sup>ε</sup> with the samples having ε<sup>0</sup> <sup>2</sup> ¼ 2.07 and ε<sup>0</sup> <sup>2</sup> ¼ 2.61 in the absence of losses (tanδ ¼ 0). The Q-factor Qε, caused by losses in the dielectric is designated by the curve 4, for the sample having parameters: ε<sup>0</sup> <sup>2</sup> ¼ 2.61 and tan<sup>δ</sup> <sup>¼</sup> <sup>10</sup>�<sup>2</sup> .

From the presented diagram one can see that behavior of the Q-factors Q<sup>ε</sup> and QR<sup>ε</sup> is quasiperiodic with the increase of the thickness hd of the samples. Such behavior, as shown earlier, is related to distribution of the electric component of the standing wave in the resonator in the plane of the top of the samples located on the bottom of the cylindrical part. The ohmic Q-factor QR<sup>0</sup> of the "empty" resonator (hd ¼ 0) is equal to 48,005. As one can see in Figure 3b, the Q-factor QR<sup>ε</sup> can essentially differ from the Q-factor QR0. For example, for the sample with ε0 <sup>2</sup> <sup>¼</sup> 2.07 and tan<sup>δ</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> at hd <sup>¼</sup> 2.48 mm, QR<sup>ε</sup> <sup>¼</sup> 33,493, that is, the ratio QR0=QR<sup>ε</sup> <sup>¼</sup> 1.43. Therefore, in order to understand how it can affect the calculation of tanδ, we write down an expression, determining its natural Q-factor Q<sup>0</sup><sup>ε</sup> of the cavity with the sample, which expresses as 1=Q<sup>0</sup><sup>ε</sup> ¼ 1=Q<sup>ε</sup> ð Þþ 1=QR<sup>ε</sup> ð Þ. For samples with low losses, the Q-factor Q<sup>0</sup><sup>ε</sup> is entirely determined by the resonator ohmic Q-factor QRε, since at the same time, Q<sup>ε</sup> >> QR<sup>ε</sup> (Figure 3b, curves 1 and 2). Hence, in the calculation of the resonator Q-factor, it is inadmissible to replace the ohmic Q-factor of the resonator with the sample by the ohmic Q-factor of the <sup>00</sup>empty<sup>00</sup> resonator, as it can result in significant errors in the valuation of tanδ for the samples having low losses (< 10�<sup>4</sup> ). Obtained conclusion is well in agreement with the results in [7].

In the investigation of the samples with high losses, a Q-factor of the resonator Q<sup>0</sup><sup>ε</sup> will already be determined by the Q-factor Qε, since here Q<sup>ε</sup> << QR<sup>ε</sup> (Figure 3b, curves 3 and 4). If the sample thickness hd < 0.25 mm, then, its placement on the bottom of the resonator cylindrical part can also lead to an incorrect result in measurement of dielectric samples losses, since in this case Q<sup>ε</sup> >> QRε. Just noted means that the natural Q-factor of the resonator should be defined by the Q-factor QRε, instead of Q<sup>ε</sup> (Figure 3b, curves 3 and 4). Hence, at the measurement of losses in thin samples ð Þ hd << λ , they should be located not on the bottom of the cylindrical part, but at the antinode of the standing wave electric field in the resonator [7].

Here, it should be noted that in the Q-factor calculations, the conductivity of the cylindrical part surface of the considered cavity was as for aluminum, and of the spherical mirror was as for brass (Figure 1). Choice of the metals depends upon the necessity to carry out an evaluation of the losses in the samples of the known dielectrics using the hemispherical ОR with the segment of the oversized circular waveguide, the mirrors of which are made of the specified metals. Conical cavity surface is considered as a perfectly conducting one.

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

Dependencies in Figure 3a corresponding to the sample thickness provide defining a real part

As an important characteristic, needed for valuation of the loss-angle tangent for dielectric samples, is Q-factor Qε, caused by losses in a dielectric (Eq. (7)), and an ohmic Q-factor QR<sup>ε</sup> of the resonator with (Eq. (9)). Dependencies of Q<sup>ε</sup> (solid curves) and QR<sup>ε</sup> (dotted curves) on the beads' thickness hd calculated using the developed theoretical model are presented in Figure 3b. The Q-factor Q<sup>ε</sup> of the cavity with the sample of the shape of a bead, permittivity

2 and 3 correspond to the ohmic Q-factor of the resonator QR<sup>ε</sup> with the samples having ε<sup>0</sup>

From the presented diagram one can see that behavior of the Q-factors Q<sup>ε</sup> and QR<sup>ε</sup> is quasiperiodic with the increase of the thickness hd of the samples. Such behavior, as shown earlier, is related to distribution of the electric component of the standing wave in the resonator in the plane of the top of the samples located on the bottom of the cylindrical part. The ohmic Q-factor QR<sup>0</sup> of the "empty" resonator (hd ¼ 0) is equal to 48,005. As one can see in Figure 3b, the Q-factor QR<sup>ε</sup> can essentially differ from the Q-factor QR0. For example, for the sample with

<sup>2</sup> <sup>¼</sup> 2.07 and tan<sup>δ</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> at hd <sup>¼</sup> 2.48 mm, QR<sup>ε</sup> <sup>¼</sup> 33,493, that is, the ratio QR0=QR<sup>ε</sup> <sup>¼</sup> 1.43. Therefore, in order to understand how it can affect the calculation of tanδ, we write down an expression, determining its natural Q-factor Q<sup>0</sup><sup>ε</sup> of the cavity with the sample, which expresses as 1=Q<sup>0</sup><sup>ε</sup> ¼ 1=Q<sup>ε</sup> ð Þþ 1=QR<sup>ε</sup> ð Þ. For samples with low losses, the Q-factor Q<sup>0</sup><sup>ε</sup> is entirely determined by the resonator ohmic Q-factor QRε, since at the same time, Q<sup>ε</sup> >> QR<sup>ε</sup> (Figure 3b, curves 1 and 2). Hence, in the calculation of the resonator Q-factor, it is inadmissible to replace the ohmic Q-factor of the resonator with the sample by the ohmic Q-factor of the <sup>00</sup>empty<sup>00</sup> resonator, as it can result in significant errors in the valuation of tanδ for the samples having

). Obtained conclusion is well in agreement with the results in [7].

In the investigation of the samples with high losses, a Q-factor of the resonator Q<sup>0</sup><sup>ε</sup> will already be determined by the Q-factor Qε, since here Q<sup>ε</sup> << QR<sup>ε</sup> (Figure 3b, curves 3 and 4). If the sample thickness hd < 0.25 mm, then, its placement on the bottom of the resonator cylindrical part can also lead to an incorrect result in measurement of dielectric samples losses, since in this case Q<sup>ε</sup> >> QRε. Just noted means that the natural Q-factor of the resonator should be defined by the Q-factor QRε, instead of Q<sup>ε</sup> (Figure 3b, curves 3 and 4). Hence, at the measurement of losses in thin samples ð Þ hd << λ , they should be located not on the bottom of the cylindrical part, but at the antinode of the standing wave electric field in the resonator [7].

Here, it should be noted that in the Q-factor calculations, the conductivity of the cylindrical part surface of the considered cavity was as for aluminum, and of the spherical mirror was as for brass (Figure 1). Choice of the metals depends upon the necessity to carry out an evaluation of the losses in the samples of the known dielectrics using the hemispherical ОR with the segment of the oversized circular waveguide, the mirrors of which are made of the specified

metals. Conical cavity surface is considered as a perfectly conducting one.

dielectric is designated by the curve 4, for the sample having parameters: ε<sup>0</sup>

<sup>2</sup> ¼ 2.61 in the absence of losses (tanδ ¼ 0). The Q-factor Qε, caused by losses in the

, is designated by the curve 1. The curves

<sup>2</sup> ¼

<sup>2</sup> ¼ 2.61 and

<sup>2</sup> by resonant frequency shift.

<sup>2</sup> <sup>¼</sup> 2.07 and loss-angle tangent tan<sup>δ</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup>

34 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

of the samples permittivity ε<sup>0</sup>

of which ε<sup>0</sup>

2.07 and ε<sup>0</sup>

tan<sup>δ</sup> <sup>¼</sup> <sup>10</sup>�<sup>2</sup>

ε0

.

low losses (< 10�<sup>4</sup>

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 3λ<sup>w</sup> (λw-waveguide wavelength). On the surface of the spherical mirror of curvature radius R ¼ 39 mm, two coupling slots are located, through one of which the signal with 1 kHz amplitude modulation from the high-frequency generator G4–142 inputs into the resonant volume, and through the second one, signal outputs to the load. The slots are tapering from the main section of the rectangular waveguide 3.6 � 1.8 mm into the narrow one 3.6 � 0.14 mm. Both coupling elements are oriented in such a manner that the vector of TE10 mode electric field in the rectangular waveguides is orthogonal to the plane of Figure 4.

Figure 4. Block diagram of the experimental setup.

The distance s from the resonator axis, on which coupling elements are located, is defined by the peak value of the electric field of ТЕМ01<sup>q</sup> mode in the plane of the spherical mirror and is equal to 5.5 mm (Figure 4). In this case ТЕ01q 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 j j¼ <sup>Г</sup> <sup>10</sup>�А=<sup>20</sup> [16]. Here, <sup>А</sup> is the difference in dB between data from the measuring polarizing attenuator 4 at the arrangement of the shorting plug in the coupling element plane and at the point of resonance. A resonant wavemeter 5 is included in the transmission line for more accurate measurement of the generator G4–142 frequency. A double-stub matcher 9 is placed into the branch of the directed coupler 3, containing a matched load 10, enabling compensation of the possible reflected from the waveguide junction signals, which can affect the accuracy of the resonance tuning. The signal, which passed through the ОR, inputs into the receiving transmission line, consisting of a measuring polarizing attenuator 11, a measuring line 12, a detector 6, a resonant amplifier 7 and an oscillograph 8. The signal, proportional to the amplitude of the standing wave voltage in the waveguide, is registered by the measuring line 12 and enters into the receiving transmission line, consisting of a crystal detector 6, a resonant amplifier 7 and oscillograph 8. VSWR of the studied OR is calculated using the formula VSWR <sup>¼</sup> <sup>10</sup><sup>B</sup>=20. Here, <sup>B</sup> is the difference in dB between the maximum and minimum attenuation values of the polarizing attenuator 11 during the probe movement along the waveguide. The photo of the experimental unit and the OR is given in Figure 5.

perturbation technique [13]. In the next step, the plunge 16 is moved inside the oversized circular waveguide 15 on the distance h ¼ 3λ<sup>w</sup> ¼ 13:172 mm (calculated value of h ¼ 3λ<sup>w</sup> ¼ 13:155 mm). Availability of the oversized circular waveguide segment results in the fact that the ТЕМ<sup>0110</sup> mode of the hemispherical ОR is being transformed into the axial-symmetric mode ТЕ<sup>0116</sup> [3]. In that case, the resonant distance is being measured already between the

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

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37

Figure 5. The photo of the OR with the segment of circular waveguide and the experimental unit.

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 f ¼ 68.667 GHz, which is measured by the wavemeter 5, again ТЕ<sup>0116</sup> 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<sup>0116</sup> mode. In order

<sup>2</sup> in that case, we use the curve 2 (Figure 3a). Measurement of two samples with the

same dimensions and made of Plexiglas are carried out similarly. The mode ТЕ<sup>0116</sup> is excited but now only at the frequencies 68.051 GHz and (hd ¼ 2.99 mm) and f ¼ 67.030 GHz (hd ¼ 3.58 mm). Using the curves 1 and 2 (Figure 3a), we evaluate permittivity of the samples having

spherical ОR mirror and the plunge.

to value ε<sup>0</sup>

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 2а<sup>2</sup> ¼ 18 mm, and it was equal to the diameter of the resonator cylindrical part. Values ε<sup>0</sup> <sup>2</sup> of the samples have been measured using the described experimental setup and curves presented in Figure 3a.

We measured ε<sup>0</sup> <sup>2</sup> for two beads having the diameter equal to 18 mm and the thickness 2.99 and 3.58 mm both made of Teflon, and two beads with the same dimensions but made of Plexiglas. Originally, the plunge is installed flush with the flat mirror, and in the hemispherical ОR, the axial-asymmetric mode ТЕМ<sup>0110</sup> is excited. At the same time, the distance between the mirrors is equal to 22.139 mm (f ¼ 71.372 GHz). The modes have been identified by application of the Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies http://dx.doi.org/10.5772/intechopen.73643 37

Figure 5. The photo of the OR with the segment of circular waveguide and the experimental unit.

The distance s from the resonator axis, on which coupling elements are located, is defined by the peak value of the electric field of ТЕМ01<sup>q</sup> mode in the plane of the spherical mirror and is equal to 5.5 mm (Figure 4). In this case ТЕ01q 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

36 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

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 j j¼ <sup>Г</sup> <sup>10</sup>�А=<sup>20</sup> [16]. Here, <sup>А</sup> is the difference in dB between data from the measuring polarizing attenuator 4 at the arrangement of the shorting plug in the coupling element plane and at the point of resonance. A resonant wavemeter 5 is included in the transmission line for more accurate measurement of the generator G4–142 frequency. A double-stub matcher 9 is placed into the branch of the directed coupler 3, containing a matched load 10, enabling compensation of the possible reflected from the waveguide junction signals, which can affect the accuracy of the resonance tuning. The signal, which passed through the ОR, inputs into the receiving transmission line, consisting of a measuring polarizing attenuator 11, a measuring line 12, a detector 6, a resonant amplifier 7 and an oscillograph 8. The signal, proportional to the amplitude of the standing wave voltage in the waveguide, is registered by the measuring line 12 and enters into the receiving transmission line, consisting of a crystal detector 6, a resonant amplifier 7 and oscillograph 8. VSWR of the studied OR is calculated using the formula VSWR <sup>¼</sup> <sup>10</sup><sup>B</sup>=20. Here, <sup>B</sup> is the difference in dB between the maximum and minimum attenuation values of the polarizing attenuator 11 during the probe movement along the waveguide. The photo of the experimental unit and the

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 2а<sup>2</sup> ¼ 18 mm, and it was equal to the

3.58 mm both made of Teflon, and two beads with the same dimensions but made of Plexiglas. Originally, the plunge is installed flush with the flat mirror, and in the hemispherical ОR, the axial-asymmetric mode ТЕМ<sup>0110</sup> is excited. At the same time, the distance between the mirrors is equal to 22.139 mm (f ¼ 71.372 GHz). The modes have been identified by application of the

<sup>2</sup> for two beads having the diameter equal to 18 mm and the thickness 2.99 and

<sup>2</sup> of the samples have been measured using

11 mm from it.

OR is given in Figure 5.

We measured ε<sup>0</sup>

diameter of the resonator cylindrical part. Values ε<sup>0</sup>

the described experimental setup and curves presented in Figure 3a.

perturbation technique [13]. In the next step, the plunge 16 is moved inside the oversized circular waveguide 15 on the distance h ¼ 3λ<sup>w</sup> ¼ 13:172 mm (calculated value of h ¼ 3λ<sup>w</sup> ¼ 13:155 mm). Availability of the oversized circular waveguide segment results in the fact that the ТЕМ<sup>0110</sup> mode of the hemispherical ОR is being transformed into the axial-symmetric mode ТЕ<sup>0116</sup> [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 f ¼ 68.667 GHz, which is measured by the wavemeter 5, again ТЕ<sup>0116</sup> 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<sup>0116</sup> mode. In order to value ε<sup>0</sup> <sup>2</sup> in that case, we use the curve 2 (Figure 3a). Measurement of two samples with the same dimensions and made of Plexiglas are carried out similarly. The mode ТЕ<sup>0116</sup> is excited but now only at the frequencies 68.051 GHz and (hd ¼ 2.99 mm) and f ¼ 67.030 GHz (hd ¼ 3.58 mm). Using the curves 1 and 2 (Figure 3a), we evaluate permittivity of the samples having


the same axial-symmetric mode TE0116 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

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

factor on the electric field, both in cavity formed by cylindrical, conical and spherical surfaces, and in cylindrical, one almost coincide. In turn, it confirms the conclusion, noted earlier, that the patterns of the electromagnetic field both in the ОR with the segment of the oversized circular waveguide and in the cylindrical cavity of same length l and diameter 2а<sup>2</sup> (Figure 1)

Now let us evaluate losses in the bead having diameter 18 mm and hd ¼ 2:99 mm thickness

of the ОR, the length of which h = 13.172 mm. Measurements are carried out using the experimental setup, a block diagram of which is presented in Figure 4, and its photo is shown in Figure 5. Eigen Q-factor of the resonant system with the measured sample Q<sup>0</sup><sup>ε</sup> is defined by

where QR<sup>ε</sup> is an ohmic Q-factor of the resonator with the sample; Qrad<sup>ε</sup> is a diffraction Q-factor

In the absence of the measured sample in the resonant volume, unloaded Q<sup>00</sup> of the ОR with

where QR<sup>0</sup> is an ohmic Q-factor of the resonator without the sample; Qrad<sup>0</sup> is a diffraction Q-

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

QR<sup>ε</sup>

Diffraction Q-factors, which are rather difficult to measure, enter into (Eq. (12)). The resonant system under consideration (Figures 4 and 5) allows to do the assumptions, which provide definition of tg δ without measuring Qrad<sup>ε</sup> and Qrad0. Since the sample is placed into the resonator cylindrical part, then it should not disturb strongly the electromagnetic field in the open part of the ОR (Figure 5a and b). Therefore, we can assume that the diffraction Q-factor of the resonator with the sample is equal to the diffraction Q-factor of the <sup>00</sup>empty<sup>00</sup> resonator,

� <sup>1</sup> QR<sup>0</sup> � <sup>1</sup> Qrad<sup>ε</sup>

: (12)

� <sup>1</sup> Qrad<sup>0</sup>

<sup>¼</sup> <sup>1</sup> QR<sup>0</sup> þ 1 Qrad<sup>0</sup>

1 Q<sup>0</sup><sup>ε</sup> <sup>¼</sup> <sup>1</sup> QR<sup>ε</sup> þ 1 Qrad<sup>ε</sup>

of the ОR with the sample caused by the coupling of the mode with free space.

1 Q<sup>00</sup>

<sup>2</sup> ¼ 2:07). The sample is placed on the plunge located in the waveguide part

<sup>2</sup> change, the behavior of the resonators energy filling

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39

þ KE tanδ, (10)

, (11)

various thickness in whole range of the ε<sup>0</sup>

the same mode can be presented as follows:

factor of the ОR without the sample.

tan<sup>δ</sup> <sup>¼</sup> <sup>1</sup> KE

1 Q<sup>0</sup><sup>ε</sup>

� 1 <sup>Q</sup><sup>00</sup> � <sup>1</sup>

losses tangent in the sample.

are identical.

made of Teflon (ε<sup>0</sup>

the relation [7, 17]

Table 1. Measured values of ε<sup>0</sup> <sup>2</sup> for beads made of Teflon and Plexiglas.

various thickness and made of Plexiglas. Frequency measurement error in the millimeter range using a resonant wavemeter is �0.05% [13]. Taking into account the frequency valuation by using the calibration curves of the resonant wavemeter, the total error of the samples permittivity measurement by application of the considered resonant cell is about ≈1%. The measurements results of the permittivity real part for considered samples are shown in Table 1.

In Table 1, Δf designates the frequency shift of TE0116 mode when putting the sample into <sup>00</sup>empty<sup>00</sup> 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 KE [7]. For its calculation, we use (Eq. (8)). The advantage of that expression is in the possibility to find the energy factor KE not by the calculations of stored energies W1<sup>E</sup> and W2<sup>E</sup> in the volumes Vð Þ<sup>2</sup> and Vð Þ<sup>1</sup> (Figure 1), but by differentiation of the resonant circular frequency ω<sup>0</sup> ε<sup>0</sup> 2 dependence, presented in Figure 3a.

The behavior KE ¼ ψ ε<sup>0</sup> 2 for the samples having the shape of beads and diameter 18 mm made of Teflon (ε<sup>0</sup> <sup>2</sup> ¼ 2:07) located on the bottom of the cylindrical part of the electrodynamic model of the ОR and having various thickness hd is shown in Figure 6 as an example.

In calculations, we assume that there is nondegenerate mode TE0116 exists in the considered resonator. In Figure 6, also by dotted shows the factor KE for a cylindrical resonator, in which

Figure 6. Dependencies of the energy filling factor of the resonator KE on permittivity of the samples having various thicknesses.

the same axial-symmetric mode TE0116 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 ε<sup>0</sup> <sup>2</sup> change, the behavior of the resonators energy filling factor on the electric field, both in cavity formed by cylindrical, conical and spherical surfaces, and in cylindrical, one almost coincide. In turn, it confirms the conclusion, noted earlier, that the patterns of the electromagnetic field both in the ОR with the segment of the oversized circular waveguide and in the cylindrical cavity of same length l and diameter 2а<sup>2</sup> (Figure 1) are identical.

Now let us evaluate losses in the bead having diameter 18 mm and hd ¼ 2:99 mm thickness made of Teflon (ε<sup>0</sup> <sup>2</sup> ¼ 2:07). The sample is placed on the plunge located in the waveguide part of the ОR, the length of which h = 13.172 mm. Measurements are carried out using the experimental setup, a block diagram of which is presented in Figure 4, and its photo is shown in Figure 5. Eigen Q-factor of the resonant system with the measured sample Q<sup>0</sup><sup>ε</sup> is defined by the relation [7, 17]

various thickness and made of Plexiglas. Frequency measurement error in the millimeter range using a resonant wavemeter is �0.05% [13]. Taking into account the frequency valuation by using the calibration curves of the resonant wavemeter, the total error of the samples permittivity measurement by application of the considered resonant cell is about ≈1%. The measurements results of the permittivity real part for considered samples are shown in Table 1.

<sup>2</sup> for beads made of Teflon and Plexiglas.

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, Δf designates the frequency shift of TE0116 mode when putting the sample into

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 KE [7]. For its calculation, we use (Eq. (8)). The advantage of that expression is in the possibility to find the energy factor KE not by the calculations of stored energies W1<sup>E</sup> and W2<sup>E</sup> in the volumes Vð Þ<sup>2</sup> and Vð Þ<sup>1</sup> (Figure 1), but by

model of the ОR and having various thickness hd is shown in Figure 6 as an example.

In calculations, we assume that there is nondegenerate mode TE0116 exists in the considered resonator. In Figure 6, also by dotted shows the factor KE for a cylindrical resonator, in which

Figure 6. Dependencies of the energy filling factor of the resonator KE on permittivity of the samples having various

2

for the samples having the shape of beads and diameter 18 mm

<sup>2</sup> Literary value ε0 2

The difference Δf, GHz

<sup>2</sup> ¼ 2:07) located on the bottom of the cylindrical part of the electrodynamic

dependence, presented in Figure 3a.

differentiation of the resonant circular frequency ω<sup>0</sup> ε<sup>0</sup>

Material Thickness of the sample hd (mm) The measured value ε<sup>0</sup>

38 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

2

<sup>00</sup>empty<sup>00</sup> ОR.

The behavior KE ¼ ψ ε<sup>0</sup>

Table 1. Measured values of ε<sup>0</sup>

made of Teflon (ε<sup>0</sup>

thicknesses.

$$\frac{1}{Q\_{0\varepsilon}} = \frac{1}{Q\_{Re}} + \frac{1}{Q\_{rad\varepsilon}} + K\_E \tan \delta,\tag{10}$$

where QR<sup>ε</sup> is an ohmic Q-factor of the resonator with the sample; Qrad<sup>ε</sup> is a diffraction Q-factor of the ОR with the sample caused by the coupling of the mode with free space.

In the absence of the measured sample in the resonant volume, unloaded Q<sup>00</sup> of the ОR with the same mode can be presented as follows:

$$\frac{1}{Q\_{00}} = \frac{1}{Q\_{R0}} + \frac{1}{Q\_{rad0}},\tag{11}$$

where QR<sup>0</sup> is an ohmic Q-factor of the resonator without the sample; Qrad<sup>0</sup> is a diffraction Qfactor of the ОR without the sample.

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

$$\tan \delta = \frac{1}{K\_E} \left[ \left( \frac{1}{Q\_{0\epsilon}} - \frac{1}{Q\_{00}} \right) - \left( \frac{1}{Q\_{Re}} - \frac{1}{Q\_{R0}} \right) - \left( \frac{1}{Q\_{nd\epsilon}} - \frac{1}{Q\_{nd0}} \right) \right]. \tag{12}$$

Diffraction Q-factors, which are rather difficult to measure, enter into (Eq. (12)). The resonant system under consideration (Figures 4 and 5) allows to do the assumptions, which provide definition of tg δ without measuring Qrad<sup>ε</sup> and Qrad0. Since the sample is placed into the resonator cylindrical part, then it should not disturb strongly the electromagnetic field in the open part of the ОR (Figure 5a and b). Therefore, we can assume that the diffraction Q-factor of the resonator with the sample is equal to the diffraction Q-factor of the <sup>00</sup>empty<sup>00</sup> resonator, that is, Qrad<sup>ε</sup> ≈ Qrad0. Taking this into account, we write down in the final form an expression determining the tangent of dielectric losses in the sample as follows:

$$\tan \delta = \frac{1}{K\_E} \left[ \left( \frac{1}{Q\_{0\epsilon}} - \frac{1}{Q\_{00}} \right) - \left( \frac{1}{Q\_{R\epsilon}} - \frac{1}{Q\_{R0}} \right) \right]. \tag{13}$$

For the samples under studies having the shape of the bead (2а<sup>2</sup> ¼ 18 mm, hd ¼ 2.99 mm) made of Teflon and Plexiglas, Q-factors Q<sup>0</sup><sup>ε</sup> and Q<sup>00</sup> are evaluated from the experiment, energy filling factor of the ОR on the electric field KE is calculated (Figure 6a), QR<sup>ε</sup> and QR<sup>0</sup> are also calculated theoretically (Figure 3b, curve 2, 3).

In order to find tanδ of the samples, loaded Q-factors QL<sup>ε</sup> and QL<sup>0</sup> of the resonator with the sample and without were measured in the case of ТЕ<sup>0116</sup> mode excitation. At the same time, loaded and unloaded Q-factors of the ОR are related as follows [9]:

$$\begin{cases} Q\_{L\varepsilon} = \eta Q\_{0\varepsilon} / (1 + \beta\_{1\varepsilon} + \beta\_{2\varepsilon}), \\ Q\_{L0} = \eta Q\_{00} / (1 + \beta\_1 + \beta\_2), \end{cases} \tag{14}$$

Taking into account the obtained values QR<sup>0</sup> and Q00, the results of tanδ measurements for the samples having the shape of the beads made of Teflon and Plexiglas of thickness equal to 2.99 mm are shown in Table 2. Measuring error of the losses in dielectric sample amounts (1÷3)% and is defined, mainly, by the errors of the frequency measurement using a wavemeter

Teflon 0.0886 49,339 1588 0.287 2604 (3.117 � 0.034) � <sup>10</sup>�<sup>4</sup> Plexiglas 0.0878 48,005 502 0.181 712 (1.192 � 0.035) � <sup>10</sup>�<sup>2</sup>

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

As may be seen in Table 2, measured values of tanδ for the samples made of Teflon and Plexiglas coincide with the results obtained by other authors. For material with low losses (Teflon), we got tan<sup>δ</sup> = (2.8 � 0.2) � <sup>10</sup>�<sup>4</sup> [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 tanδ differs by 1.7% from the results of other authors (tanδ = (1.1 �

basic possibility to measure electromagnetic parameters of materials with high losses in millimeter and sub-millimeter ranges by means of the resonant system consisting of the hemispherical ОR with the segment of the oversized circular waveguide. In order to increase the accuracy

<sup>2</sup> and tanδ measurements for dielectric materials with low losses, it is necessary to increase the Q-factor of the resonant system. To achieve it, the resonator with the mirrors of large

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 ε2, μ2. The rest part

We confine ourselves to the analysis of axially-symmetric oscillations of TE-mode, which in cylindrical coordinates, where axis z coincides with the resonator symmetry axis, have Eφ-, Hrand Hz- components of the electromagnetic field. For this oscillation mode, the initial problem for the Maxwell equations can be reduced to finding wave numbers k, for which there exist the

nontrivial solutions U<sup>1</sup> and U<sup>2</sup> of the two-dimensional Helmholtz equations.

<sup>2</sup> and tanδ measurements. We showed only the

http://dx.doi.org/10.5772/intechopen.73643

41

and loaded Q-factors QL<sup>ε</sup> and QL<sup>0</sup> of the resonator with the sample and without it.

Material KE Q<sup>R</sup><sup>ε</sup> Q<sup>L</sup><sup>ε</sup> βε Q<sup>0</sup><sup>ε</sup> tg δ

aperture and the spherical reflector of the larger curvature radius should be used.

0.06) � <sup>10</sup>�<sup>2</sup>

of ε<sup>0</sup>

) [20].

3.1. Resonator model

The authors did not aim to get high accuracy ε<sup>0</sup>

Table 2. Measurement results of tanδ for the studied samples.

3. Open resonator with a dielectric rod

of the resonator is assumed to be empty ε<sup>1</sup> ¼ μ<sup>1</sup> ¼ 1 (area 1).

where β<sup>1</sup><sup>ε</sup> and β<sup>1</sup> are coupling coefficients of the resonator with the receiving waveguide line with the sample on the bottom of the cylindrical part and without it; β<sup>2</sup><sup>ε</sup> and β<sup>2</sup> are coupling coefficients of the resonator with a load with the measured sample and without it, η is efficiency of excitation of the ТЕ<sup>0116</sup> 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<sup>0116</sup> 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, β<sup>1</sup><sup>ε</sup> ¼ β<sup>2</sup><sup>ε</sup> ¼ βε and β<sup>1</sup> ¼ β<sup>2</sup> ¼ β. In the paper [19], it was shown that the transmission type resonator having equal input and output coupling cannot be recoupled. It indicates that βε and β in this case are less than 1 and therefore are equal to 1=VSWR and the reflection coefficient Г > 0. Measurements of Г performed using the directional coupler (VSWR ¼ ð Þ 1 þ Г =ð Þ 1 � Г ) and of VSWR using the measuring line (Figures 4 and 5) confirmed the conclusion made of an equality of the input and output coupling. In such a way, from (Eq. (14)) we can find out the natural Q-factors of the resonant cell with the sample Q<sup>0</sup><sup>ε</sup> and without it Q<sup>00</sup> in (Eq. (13)).

$$\begin{cases} Q\_{0\varepsilon} = Q\_{L\varepsilon} \left( 1 + 2\beta\_{\varepsilon} \right) / 0.96 \,\prime \\ Q\_{00} = Q\_{L0} \left( 1 + 2\beta \right) / 0.96 \,\prime \end{cases} \tag{15}$$

For calculation of the loss-angle tangent in the samples made of Teflon and Plexiglas, we calculate the ohmic Q-factor of the <sup>00</sup>empty<sup>00</sup> resonator using (Eq. (13)). As seen from Figure 3b (curves 2, 3, hd ¼ 0), QR<sup>0</sup> ¼ 48,005. Experimentally measured loaded Q-factor QL<sup>0</sup> of the <sup>00</sup>empty<sup>00</sup> resonator, in which the mode ТЕ<sup>0116</sup> exists, turned out to be equal to 1660. At the same time, the coupling coefficient β ¼ 0.310 (Γ ¼ 0.527). Now from (Eq. (15)), we can calculate that the natural Q-factor of the <sup>00</sup>empty<sup>00</sup> resonator equals to Q<sup>00</sup> = 2801.


Table 2. Measurement results of tanδ for the studied samples.

that is, Qrad<sup>ε</sup> ≈ Qrad0. Taking this into account, we write down in the final form an expression

For the samples under studies having the shape of the bead (2а<sup>2</sup> ¼ 18 mm, hd ¼ 2.99 mm) made of Teflon and Plexiglas, Q-factors Q<sup>0</sup><sup>ε</sup> and Q<sup>00</sup> are evaluated from the experiment, energy filling factor of the ОR on the electric field KE is calculated (Figure 6a), QR<sup>ε</sup> and QR<sup>0</sup> are also

In order to find tanδ of the samples, loaded Q-factors QL<sup>ε</sup> and QL<sup>0</sup> of the resonator with the sample and without were measured in the case of ТЕ<sup>0116</sup> mode excitation. At the same time,

QL<sup>ε</sup> ¼ ηQ<sup>0</sup>ε= 1 þ β<sup>1</sup><sup>ε</sup> þ β<sup>2</sup><sup>ε</sup>

where β<sup>1</sup><sup>ε</sup> and β<sup>1</sup> are coupling coefficients of the resonator with the receiving waveguide line with the sample on the bottom of the cylindrical part and without it; β<sup>2</sup><sup>ε</sup> and β<sup>2</sup> are coupling coefficients of the resonator with a load with the measured sample and without it, η is

Based on the research carried out with a metal screen covering the ОR [18], it was determined that excitation efficiency of the considered ТЕ<sup>0116</sup> 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, β<sup>1</sup><sup>ε</sup> ¼ β<sup>2</sup><sup>ε</sup> ¼ βε and β<sup>1</sup> ¼ β<sup>2</sup> ¼ β. In the paper [19], it was shown that the transmission type resonator having equal input and output coupling cannot be recoupled. It indicates that βε and β in this case are less than 1 and therefore are equal to 1=VSWR and the reflection coefficient Г > 0. Measurements of Г performed using the directional coupler (VSWR ¼ ð Þ 1 þ Г =ð Þ 1 � Г ) and of VSWR using the measuring line (Figures 4 and 5) confirmed the conclusion made of an equality of the input and output coupling. In such a way, from (Eq. (14)) we can find out the natural Q-factors

QL<sup>0</sup> ¼ ηQ00= 1 þ β<sup>1</sup> þ β<sup>2</sup>

� <sup>1</sup> QR<sup>ε</sup>

� �,

� �,

� � � �

� <sup>1</sup> QR<sup>0</sup>

: (13)

(14)

(15)

� 1 Q<sup>00</sup>

� �

determining the tangent of dielectric losses in the sample as follows:

40 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

loaded and unloaded Q-factors of the ОR are related as follows [9]:

(

efficiency of excitation of the ТЕ<sup>0116</sup> mode in the considered resonator.

of the resonant cell with the sample Q<sup>0</sup><sup>ε</sup> and without it Q<sup>00</sup> in (Eq. (13)).

that the natural Q-factor of the <sup>00</sup>empty<sup>00</sup> resonator equals to Q<sup>00</sup> = 2801.

(

Q<sup>0</sup><sup>ε</sup> ¼ QL<sup>ε</sup> 1 þ 2βε

<sup>Q</sup><sup>00</sup> <sup>¼</sup> QL<sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>β</sup> � �=0:96: :

For calculation of the loss-angle tangent in the samples made of Teflon and Plexiglas, we calculate the ohmic Q-factor of the <sup>00</sup>empty<sup>00</sup> resonator using (Eq. (13)). As seen from Figure 3b (curves 2, 3, hd ¼ 0), QR<sup>0</sup> ¼ 48,005. Experimentally measured loaded Q-factor QL<sup>0</sup> of the <sup>00</sup>empty<sup>00</sup> resonator, in which the mode ТЕ<sup>0116</sup> exists, turned out to be equal to 1660. At the same time, the coupling coefficient β ¼ 0.310 (Γ ¼ 0.527). Now from (Eq. (15)), we can calculate

� �=0:96,

1 Q<sup>0</sup><sup>ε</sup>

tan<sup>δ</sup> <sup>¼</sup> <sup>1</sup> KE

calculated theoretically (Figure 3b, curve 2, 3).

Taking into account the obtained values QR<sup>0</sup> and Q00, the results of tanδ measurements for the samples having the shape of the beads made of Teflon and Plexiglas of thickness equal to 2.99 mm are shown in Table 2. Measuring error of the losses in dielectric sample amounts (1÷3)% and is defined, mainly, by the errors of the frequency measurement using a wavemeter and loaded Q-factors QL<sup>ε</sup> and QL<sup>0</sup> of the resonator with the sample and without it.

As may be seen in Table 2, measured values of tanδ for the samples made of Teflon and Plexiglas coincide with the results obtained by other authors. For material with low losses (Teflon), we got tan<sup>δ</sup> = (2.8 � 0.2) � <sup>10</sup>�<sup>4</sup> [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 tanδ differs by 1.7% from the results of other authors (tanδ = (1.1 � 0.06) � <sup>10</sup>�<sup>2</sup> ) [20].

The authors did not aim to get high accuracy ε<sup>0</sup> <sup>2</sup> and tanδ measurements. We showed only the basic possibility to measure electromagnetic parameters of materials with high losses in millimeter and sub-millimeter ranges by means of the resonant system consisting of the hemispherical ОR with the segment of the oversized circular waveguide. In order to increase the accuracy of ε<sup>0</sup> <sup>2</sup> and tanδ measurements for dielectric materials with low losses, it is necessary to increase the Q-factor of the resonant system. To achieve it, the resonator with the mirrors of large aperture and the spherical reflector of the larger curvature radius should be used.
