Coupling Coefficients of Different Disk Microresonators with Whispering Gallery Modes

The coupling coefficients of the disk microresonators, consisting of different dielectrics, are presented. The analytical expressions for the coupling coefficients are obtained. The basic regularities of the coupling coefficient changing on the structure parameter variations are considered. The coupling coefficients as functions of main structure parameters are studied. The calculation results of the transmission coefficients as well as the reflection coefficients of the bandpass filters, building up on different disk microresonators with different modes are presented. The S-matrix frequency dependences of the bandpass filters on different disk microresonators in the infrared wavelength range are calculated. The dielectric Q-factor values, necessary for acceptable measure of the scattering are determined. The amplitude-frequency characteristics of the bandpass filters on the vertically coupled as well as laterally coupled disc microresonators are investigated. Most optimal configurations of coupled microresonators, allowing to achieve the best scattering characteristics are determined. It’s showed that using of microresonators with whispering gallery modes allow relatively easy obtain frequency-symmetrical characteristics of the filters.


Introduction
Disk dielectric microresonators with whispering gallery (WG) modes inscribes in the planar integral circuits quite naturally. Today, ones are being actively studied for purpose of their application in the different devices of the optical, infrared and terahertz wavelength ranges [1][2][3][4][5][6][7]. Eigenoscillations of two disk dielectric microresonators were considered in [4][5][6][7], but its coupling coefficients were not calculated and not studied in full measure. For calculation and optimization of the device parameters, it's convenient to carry out on basis of electrodynamic modeling with using coupling coefficients [8].
The goal of the present work is the analytical calculation of the coupling coefficients of different disk microresonators with WG modes in the open space. In this article, we also use the scattering theory for S-matrix coefficients calculation of different microresonator bandpass filters.
1 Eigenoscillation field calculation of the disk microresonator For the coupling coefficients calculation, information about microresonator eigenoscillation fields is necessary. Most simple analytical presentation the field within dielectric cylinder can be obtained in the form of so-called one-wave approximation [9]. Toward this end, an electromagnetic field is written in the cylindrical coordinate system ( , , ) (see fig. 1a), located in the microresonator center, approximately in the form of hybrid standing wave of the circular dielectric waveguide section: Here 1 , ℎ 1 -is the electrical and magnetic field amplitudes; , -is the wave numbers; 0 -is the radius; -is the height of the disk microresonator; ′ ( ) -is the derivative of Bessel function of the first kind of the -th order [10]. The constant values 1 , ℎ 1 ; , approximate expression can be determined from the simple equations: 1 = 0 ; ℎ 1 = ℎ 0 and The dimensionless parameters: ⊥ = 0 ; 0⊥ = 0 0 , and also = /2; 0 = 0 /2 can be calculated from combined equations: ; as well as, for the ± and ∓ modes: ′ ( ) -is the derivative of Bessel function of the second kind of the -th order.
The defines the number of half-waves, located in the radial direction inside dielectric cylinder, the defines half-waves number, located in the direction of -axis in the microresonator material; the is integer. Sign +(−) in the cases of ± mode corresponds to an even (odd) mode distribution of -component of the magnetic field; and the −(+) for the ∓ mode corresponds to an even (odd) mode distribution ofcomponent of the electric field in the microresonator (see fig.1) relative to the plane of symmetry = 0 ( fig. 1a).
2 Calculation of coupling coefficient The fields and frequencies of several microresonator coupled oscillations defines by values of the coupling coefficients. In the common case, the coupling coefficient can be determined as a surface integral: expressed via the eigenmode field (⃗ , ⃗ ℎ ) of one (th) microresonator on the surface of another ( -th) microresonator. Here , = 1, 2; and ⃗ -is the normal to the surface of -th microresonator, 0 -is the resonance frequency; -is the energy, stored in the dielectric of -th microresonator.
The real and imaginary parts of the coupling coefficient can take positive values as well as negative ones. As follow from (5), the coupling coefficients of two different microresonators are not equal to one another: The integral (5) can be calculated, based on early known analytical expression for the coupling coefficients, of the Cylindrical DRs in the Rectangular metal waveguide. In this case, required analytical expressions for mutual coupling coefficients 12 , can be received by transferring the waveguide walls to the infinity. Using necessary expressions, after simplifications, we obtain: In the case of two different microresonators, with the same parity of each field on and on the indexes ( = 1, 2 ), relative to the plane of symmetry: − = 0 ( = 1, 2) (see fig. 2a), the mutual coupling coefficients can be obtained in the form: in the area: where the top sign of (6) corresponds to even-mode field distribution, relative to the plane of symmetry and the bottom one corresponds to odd-mode field distribution (see 1) -is the height; -relative dielectric permittivity of the -th disk microresonator ( = 1, 2); ; ℎ 0 -are the normalized amplitudes, defined from (2); In the case of different disk microresonators with equal parity of the field distribution (see fig. 3a) relative to symmetry plane ′ ( fig. 1b): in the area: ∆ > 1 /2 + 2 /2 obtains where sin ∆ = ∆ /∆ ; ∆ = √︀ ∆ 2 + ∆ 2 ; Рис. 3. Vertically coupled different disk microresonators (a). Coupling coefficient dependencies for the  (10), as well as for (7).
The coupling between disk microresonator and the open space following from Helmholtz-Kirchhoff's integral theorem can be obtained as: 3 Coupling coefficient analysis Discovered relationships valid for any eigenoscillations of the disk microresonators in the open space, but a greatest interest presents the WG modes, as it's well known, possessed a highest possible quality.  . 2b -e;  fig. 3b -c; fig. 4). Increase of the distance between resonator centers is accompanied by rapid coupling coefficient decrease. According to that, relative motion in the tangent direction leads to complex interference ( fig. 2b, d; fig. 3b; fig. 4a, c) of their mutual influence, determining by significant eigenmode field variation nearby their surface.
The greatest amount of coupling between microresonators appears on its coaxial arrangement ( fig. 3b, c). The sign of the coupling coefficients extreme values is determined both by the azimuth numbers and mutual microresonators position ( fig. 2c, e).
Imaginary parts of the coupling coefficients are more smooth functions on coordinates ( fig. 2b, d;  fig. 3b; fig. 4a, c). For selected dielectric permittivity, its values are approximately one tenth degrees to the real parts.
The fig. 2b is showing difference between two resonator coupling coefficients.

Filter parameters calculation
Obtained results allow us to create electrodynamic models of various filters in the millimeter, terahertz or infrared wavelength ranges. As can be seen the coupling between not adjacent microresonators in the filters will be very small in comparison with coupling between adjacent ones, that allows simply to build filters with symmetrically parameters of the scattering. Different microresonator using in this case will allow to obtain much clean stop-bands.
The transmission T and the reflection coefficient of different DR system for the bandpass filter configuration in the transmission line can be obtained by using perturbation theory:      Here for different microresonators, the functions ( ) are dependent on the partial DR and the coupled oscillation numbers: -is the loss power in the dielectric of -th DR; -is the complex amplitude of -th resonator of the -th modẽ frequency of the filter [8].
The fig. 5 and 6 show results of the calculation of bandpass filter -parameters matrix, that is built up on different disk microresonators with It's seen, that in consequence of rapidly coupling coefficients decrease, all -parameters are symmetrical functions on the frequency. As we used a large number of resonators, the 21 squareness was obtained well.
Рис. 5. Sketch of bandpass filter on laterally coupled different disk microresonators (a). -matrix responses of the 7-section bandpass filter on

Conclusions
Analytical relationships for the coupling coefficients of the different disk microresonators in the Open space have been obtained and investigated.
It's stated, that the coupling between not adjacent microresonators in the filters is small in comparison with coupling between adjacent ones that allows simply building bandpass filters with symmetrically scattering parameters.
The filters on WG modes microresonators has acceptable frequency responses and after optimization maybe recommended for utilization on multiplexing of various communication systems.