Scattering of Electromagnetic Waves on Different Dielectric Resonators of the Microwave Filters

The theory of scattering of electromagnetic waves by systems of various coupled the different shape and variant permittivity dielectric resonators is expanded. A new definition of the coupling coefficients of different dielectric resonators is given. The analytical expressions of coupling coefficient of different cylindrical and spherical dielectric resonators made from different dielectrics are obtained. The main regularities of the change in the coupling with the variation of the structure’s parameters are considered. The results of calculation of the transmission and the reflection coefficients for bandpass and bandstop filters on various dielectric resonators in the rectangular and circular waveguides are presented. Most optimal configurations, allowing to achieve the best scattering characteristics are determined.


Introduction
It is well-known that in addition to a very high factors the dielectric resonators (DR) have a number of disadvantages, such as increased density of the spectrum, in some cases non-optimal coupling. The scattering parameters of a variety of devices can be significantly improved by using different forms of DR [1][2][3][4][5][6][7][8]. For the purposes of theory development all resonators are generally supposed to have the same shape and manufactured of the same dielectric [9]. In order to improve the parameters in some cases there is need to build filters on DRs with different shapes made of variant dielectrics. However, in this case the theory describing scattering processes becomes more complicated. In this article we developed electrodynamic theory for describing different DRs. The equation system for the unknown amplitudes of the DR coupled oscillations have been obtained. Total analytical solutions have been found. The research of electromagnetic wave scattering on different cylindrical and spherical DR structures have been conducted in the propagating waveguide and evanescent waveguide segment. In special case of identical resonators obtained solutions are simplified to the known ones [9].

Statement of the problem
The goal of the current article is the development of the theory of microwave filters, consisting of different DRs, that can be used in the modern communication systems.
2 Scattering theory Consider the system of different DRs, consisting of different materials. We assume that the eigenoscillation field of each isolated DR is known: (⃗ , ⃗ ℎ ), ( = 1, 2, ..., ). Here ⃗ -is the electric field and ⃗ ℎis the magnetic field intensity of the -th isolated DR. The eigenoscillation field of the -DR system (⃗, ⃗ ℎ) can be found as a superposition of fields of the isolated resonators: As shown, the DR amplitudes should satisfy the equation system [10]: is the real part of frequencies of isolated partial resonators ( = 1, 2, ..., ); = Re(˜− 0 ); ′′ = Im(˜);˜-is the complex resonance frequency of the DR system; ,˜are the coupling coefficients of the -th and -th DRs on damped and expanding waves of the transmission line [10] respectively: Here -is the ultimate multi index, determining the numbers of the expanding waves in the line, and a ( ± ) 0 -is the expansion coefficient of the -th DR field on the -th wave of the transmission line [10], calculated in the coordinate system, associated with -th resonator center; ∆ = | − |; -is the longitudinal coordinate of the -th DR; Γ -is the longitudinal wave number of the transmission line; and stored in the dielectric of the -th DR; = Re˜; = − ′′ -is the complex dielectric permittivity and the 0 is the permeability of the -th DR.
In the case of different DRs the coupling coefficients (4) take different views: ̸ = and˜̸ =˜. Generally, by providing the solution to the equation systems (2) for each obtained , it is possible both to calculate approximately the complex frequency˜of the system coupling oscillation and to determine all amplitudes of partial resonators ⃗ The problem solution of the waveguide wave (︁ ⃗ + , ⃗ + )︁ scattering on the DR system will be searched in the form of expansion on coupling modes of the DR lattices (1): where (⃗ , ⃗ ℎ ) is the field of the -DR systems (1), corresponding to the eigenvalue (3). By using perturbation theory, after the volume integration of each partial resonator, the equation system with the unknown coefficients has been obtained in the form: where for different resonators, the functions ( ) are dependent on the partial DR and the coupled oscillation numbers: -is the loss power in the dielectric of -th DR. The transmission and the reflection coefficient of different DR system in the transmission line can be obtained froms (5), s (1) in the form: Here 0 , 0 are the transmission and reflection coefficients of the transmission line without DRs; , we have to calculate coupling coefficients of different DRs in the transmission line. Suppose we have two DRs of cylindrical shape with radius 1 and 2 , height of 1 and 2 , respectively. Assume that each resonator is excited in the fundamental magnetic oscillation + 101 [10]. In this case, the coupling coefficient is of the form: In the case of coupling on propagating rectangular waveguide wave 10 : -is the dielectric permittivity; 0 is the permeability and  Example of mutual coupling coefficients calculations for two different cylindrical DRs in the rectangular waveguide, obtained on a basis of (10), (11), is showed in fig. 1 a-b. As follows from (10), the difference between the values of the coupling coefficients is mainly due to the different value of stored energy in the resonator material.
The use of different spherical DRs in some cases allow to improve the filter parameters. We have represented the coupling coefficients of different spherical DRs with magnetic oscillations 111 in the form: where ( , , ) -are rectangular coordinates of theth spherical DR centers in the waveguide: ∆ = | 1 − 2 |; ( ); ( ) are the spherical Bessel and the Neumann functions, respectively [11]. The characteristic parameters = , = 0 can be obtained from the equations for natural oscillations of the -th spherical DR [12]. Here -is the radius of the -th spherical DR.
For the coupling on propagating wave 10 : The mutual coupling coefficients, calculated for two different spherical DRs in the rectangular waveguide, obtained from (12) -(14), are showed in fig. 1 c-d. As can be seen, the difference between the values of 12 and 21 in this case is small.
If two different spherical DRs located on the axis of the cylindrical metal waveguide [12], the mutual coupling coefficients becomes: Γ , -is the longitudinal wave number for the magnetic, electrical cylindrical waveguide waves, respectively; , , ( ′ , ) is the -th root of the Bessel ( ) (derivative of the Bessel ′ ( )) function [8].

Bandstop Filters on different DRs
DRs in regular transmission lines represent the most interesting case from the point of view of the theory, because in this case all the resonators at the same time exchange fluctuations both propagating and by not extending waves. DRs with 2 = 82; 2 = 1500; ∆ 2 = 2 /2 2 = 0, 8, calculated by the formula (2),(7), (8), (9) with help of the (10), (11). All resonators placed on the waveguide axis. The distance between adjacent DR centers was equal to /4, where -is the guided wavelength. The result of the scattering of the rectangular waveguide waves 10 on the structure of 9 different spherical DRs is shown in fig. 3 b -e. The coupling coefficients of the DRs were calculated by the formulas (12)-(15).
As can be seen, the use of different alternating DRs in this case gives acceptable results for the frequency distribution of scattering parameters.

Bandpass filters on different DRs
The best results were obtained for the DR structures, located in the evanescent waveguide segment and forming bandpass filters. The filters containing the DRs should have bands free of spurious oscillations. A known solution to this problem is to use different forms of DR. Fig. 4. shows scattering parameters of the filter, consisting of two lattices with different cylindrical DRs. First lattice contains 4 DR with 1 = 36, ∆ 1 = 0, 8, the second lattice consists of 7 DR with 2 = 81, ∆ 2 = 0, 4. The distance between the centers of adjacent first type DRs was equal to 17 mm; for the second type DRs 21 mm. All resonators placed on the waveguide axis.   fig. 1, fig. 6 d, e), where -is the velocity of light; | − 1 | -is the longitudinal length of the filter. As can be seen from Fig. 6 d, e, the results demonstrate a remarkable slowing of signal propagation, characteristic of the filters on DRs.
Proposed enhancement of the electrodynamic theory for the scattering electromagnetic waves on different dielectric resonators greatly enhances design of the filters and other devices. As shown from calculations, the developed model correctly describes the scattering processes in the system of different DRs for a variety of transmission lines. The obtained solutions makes it possible to calculate all scattering parameters of the filters, made in various DR. Such design has several advantages compared with identical resonators for filters, in particular the filters have a more clean stop band, and in the case of spherical cavities, produce better scattering characteristics due to the wider coupling bands variations. The frequency dependence of the scattering S-matrix can be further improved even more after a fine optimization of the filter parameters.

Conclusion
A scattering theory on different dielectric resonator systems, based on perturbation theory, has been expanded.
Given new definitions of coupling coefficients for the different dielectric resonators in the transmission line.
New analytical relationships for the coupling coefficients of different spherical and cylindrical dielectric resonators has been obtained.
A new design of the bandstop and bandpass filters on different DRs are proposed.