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Електродинаміка. ПристроїНВЧ діапазону. Антенна техніка
УДК 621.372
COUPLING COEFFICIENTS OF THE DISK DIELECTRIC MICRORESONATORS WITH WHISPERING GALLERY MODES1
Trubin A. A., Doctor of Engineering, Professor,
National Technical University of Ukraine «Kyiv Polytechnic Institute»,
Kyiv, Ukraine [email protected]
КОЕФІЦІЄНТИ ЗВ'ЯЗКУ ДИСКОВИХ ДІЕЛЕКТРИЧНИХ РЕЗОНАТОРІВ З КОЛИВАННЯМИ ШЕПОЧУЧЕЇ ГАЛЕРЕЇ
Трубін О.О., д.т.н., професор
Національний технічний університет України «Київський політехнічний інститут», Київ, Україна,
Introduction
Disk dielectric microresonators with whispering gallery (WG) modes quite naturally inscribes in the planar integral circuits. Today ones are actively studying for purpose of their application in the different devices of the optical, infrared and terahertz wavelength ranges [1 - 7]. For calculation and optimization of the device parameters it's convenient to carried out on basis of electrodynamic modeling with using coupling coefficients [8]. For that it's necessary be able to calculates microresonators both coupling coefficients with various transmission lines, and mutual coupling coefficients between its. Eigenoscillations of two disk dielectric microresonators was considered in [4 - 7], but its coupling coefficients did not calculated and not studied in full measure. The goal of the present work is the calculation and analysis of the coupling coefficients of the disk microresonators with WG modes in the Open space.
Calculation eigenoscillation field of the disk microresonator
For the coupling coefficients calculation it's necessary information about microresonator eigenoscillation fields. 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, writes an electromagnetic field in the cylindrical coordinate system (p, a,z) (see fig. 1,a), approximately in the form of hybrid standing wave of the circular dielectric waveguide section:
e JG;„(Pp) + li,m^ ■ idM
1 p „лик/ 1 p pp
f sinmal cos Pz z
[ cosmaj - sin Pz z
e
p
1 Електронний варіант статті: http://radap.kpi.ua/radiotechnique/article/view/1064
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Єа =
em ^ • І^ІМ + h ^ • J^ (Рр)
Р Рр 1 Р
ez = eiJm (РР)
Г cosma І cos Р z
[-sinmaj - sin Рz
sin та
cos та
h
h
^m^ • Jm (Рр) - h ^ • J^ (Рр)
1 Р Рр 1 Р mV '
ІЮЄі , ч , Рz Jm (РР)
-e1 1 • Jm (Рр) + him-z *—-
1 Р mV ' 1 Р Рр
hz = h1Jm (M<
cosma
- sinma
sin Р z cos Р z cosma -sinma sinma cosma I cos Р z -sinР z
sin Р z cos Р z sin Р z cos Р z
(1)
Here р ^ rc; lzl ^ L / 2, where L - is the height, and r0 - is the radius of the dielectric cylinder (fig.1, a); J'm( x) - is the derivative of Bessel function of the first kind of the m -th order; e^p - is the electrical and magnetic field amplitudes; Р, Р2 - is the wave numbers.
Fig. 1. A - sketch of the disk microresonator in the open space. The field of the coupling oscillations of two equal disk microresonators with even (b); odd (c) £H+201 modes;
(m = 20): в 1r = 9,6; A = L/2r0 = 0,2.
Approximate solution of the microresonator field in the external space can be made by various manner [1]. Depending on that, will be determined the constant values e^h, Р, Р. For a small dielectric permittivity most often uses a relationship, proportional to the Hankel function of the second kind: Н(2}(Р0р). At the same time, for the receiving of real values of characteristic parameters, the Hankel functions should be replaced to the Neumann functions Hf^) « Хі(Рр) . At that, the amplitudes e:,p convenient to presents in the standardize form: e = ae0 and h = ah0, where the e0,h0 can be determined
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Електродинаміка. ПристроїНВЧ діапазону. Антенна техніка
from more simple equations:
h + e = Jm-l(P±) + Yrn-l(P0±) • h _ e = Jm+l(P±) + Yrn+l(P0±)
0 0 P±Jm(P±) P0±Ym(P0± У 0 0 P±Jm(P±) Po±Ym(Po±)
(2)
Where dimensionless parameters: p± =p r0; p01 = P0 r0, and also pz = Pz L/2 and p0z = P0zL / 2 can be calculated from combined equations (see, for example, [1]):
~bk Jm (p г,) | і Y,m(pp г, >
P Jm (P Г, ) P0 Y]n (P0 Г, )
l4(Py)+ 1 y;,(Pq r0)
P Jm (P Г0 ) P0 Ym (P0 Г0 )
mPz
k0r0 У
1 1
—г H--:
P2 P
0 У
P2 + P2 = k2;
as well as for the HE *ml modes:
P0z
P0 + P2 = k0; P2-P2z = k
0
Pz •
-CtgPzL / 2
tgPzI. / 2
for the EH
+
nml
modes:
S1rP0z
Pz •
tgPzL/2
ctgPzL / 2_.
(3)
Here n defines a number of half-waves, located in the radial direction inside dielectric cylinder, and l defines a half-waves number, located in the direction of z -axis in the microresonator material. Sign + (-) in the cases of HE*ml mode responds to an even (odd) distribution of z -component magnetic field; and the -(+) for the EH^ml mode responds to an even (odd) distribution of z-component electric field in the microresonator (see fig. 1) relative a plane of symmetry z = 0 (fig. 1, a).
Coupling coefficient calculating
Allocation microresonators side by side with each other leads to the coupling oscillations appearance. The fields and frequencies of this oscillations defines by values of the coupling coefficients. In the common case, the coupling coefficient we determined as a surface integral:
k =
2®0wn(1 + Ssn)
(4)
i
expressed via the eigenmode field (es,hs) of one (s-th) microresonator on the surface of another (n-th) microresonator. Here s,n = 1,2; and n - is the normal to the surface sn of n -th microresonator, ш0 - is the resonance frequency; wn -is the energy, stored in the dielectric.
In special case of one isolated microresonator: (s = n) кт = iQ-1, where
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Qn is the usual quality factor.
As follow from [8], eigenoscillations of two identical microresonators take on form of even and odd spatial field distributions with respect to symmetry plane, located between ones. In this case (4): кп = к22; к12 = к21 and
b2 = +ь;; b2 = -bf; \2 = ik, ± k21 ,
responding to a cophased, or even, and antiphased, or odd, field distribution of the coupling oscillations (fig. 1, b, c). From here, using a definition [8], and determined the real and imaginary part of the frequency coupling oscillations:
/5® • g>\
к = 2 • (--v і—),
® 0 ® 0
obtain:
1,2 1
Re( —) = 1 ± — Re( к 21);
®0 2
,Л,2
Im( —)
^IX+ImOcJ],
or
5® ®
Re(K2—) = — = Re(— gv
® 0; Im(K21) =Im(®
1
1
). (5)
2 Q1 Q2)
Thus it's seen, that the real part of the mutual coupling coefficient (5) now expressed via relative frequency difference, as well as an imaginary part of the coupling coefficient is proportional to the power radiation difference of even and odd coupling modes. As follows from (5), the real and imaginary part of the coupling coefficient can take as a positive well as a negative values.
Direct calculation of the integral (4) is sufficiently complicated problem, since we don't know external fields of the disk microresonators. The integral (4) can be calculated, on a base of 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 к12 , can be received by transferring the waveguide walls to the infinity. Using necessary expressions, for example, 3. of the [10], as well as [11, 12], after simplifications obtain:
A. In the case of two identical microresonators, with the same parity of each field, relatively a plane of symmetry: y - ys = 0 (s = 1,2) (see fig. 2, a), the mutual coupling coefficients can be obtained in the form: in the area: Az > 2r0; Ax < L:
к
1,2
321 • Pz
£1rWnml Pz
(6)
•{"F©2 {н^коДрТГІЬ + со8(2тДф)Н<2:(к0Др7Г1Г)}со5(к0Дхі;)(1§,
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where the top sign of (6) responds to even field distribution, relatively the plane of symmetry and a bottom one responds to odd field distribution (see 1);
sin Дф = Az / Ар; Ap = ^Ay2 + Az2; Ax = x2 - xl; Ay = y2 - y; Az = \z2 - zl |.
Here
F(!)2 = |F,(£,)|2 + Ц,
F, (5) = Є0 {[kN/l^Jm ' ' (q i') - k0pj„ ' J (q 1')] ■
p
1
P2 - (k0 5)2
+
(pz cospz sinqz!- k0! sinpz cosqz!)
(pz sinpz cosqz!- k0! cospz sinqz!).
+[PJ„(pi K(q їлА7!2) - kJTi2J„'(p1 )Jm(q JT!2)] ■
1 Г(k sinpz cosqz!- kopz cospz sinqz!)
(k cospz sinqz!- k0pz sinpz cosqz!)
p2 - k|( 1 -52)
}■
-h0m !(k 1 . k°^ Jm(p±)Jm(q±yl1 -!2)■ p± k^/1-2
1
P2 - (k° 5)2
(pz sinpz cosqz!- k°! cospz sinqz!) (pz cospz sinqz!- k°! sinpz cosqz!)
(7)
F2(5) = (k2 - k02){e°m
1
Jm(pi )Jm(q ^Т1?2) - h°
pAVT-2 1.........^ " ° P2 - k°(1 -52)
№Jm(pi)J'm(q^/Г-!2) - k^/r-!rJm'(pi )J„(qіЛ^7?)]} ■
1 Г (pzsin pzcos qz! - k°! cos pzsin qz!)’
Pz2 - (k° !)2 (pz cospz sinqz!- k°! sinpz cosqz!)
e0; h0 - is the normalized amplitudes, defined from (2);
w
nml
(Є° - h°) [Jm-1(p± ) - Jm(p± )Jm-2(p± )1 +
0 '-'-OJ IF m-1 VF1
+(Є0 + h0 )2 [J2m+1(pi) - Jm (p1 )Jm+2 (p1 )] +
1
+2(T" h0 ) [Jm (p1 ) - Jm-1 (p1 )Jm+1 frl)]
2pz + sin2pz
2pz - sin2pz
+
+
Pz
(-7T e0 ~ f h0 ) [Jm-1 (p1 ) - Jm (p1 )Jm-2 (p1 )] +
Pz k1
z
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+А ^ и. )2 [jm.,(p±) - Jm(p± у„+2 (p±)]+
Pz k1
p
+2(— e0 ) [Jm (Pl) - Jm-1 (Pl )Jm+1 (P±)] pz
2Pz - sin2Pz 2p + sin2Pz
К К; k0 = ш0 / c; щ - is the circular resonance frequency; c - is the light velocity.
Fig. 2. Mutual coupling coefficients as a function coordinates of the microresonators with
s 1r = 9,6; A = 0,2, for EH+201 modes; (m = 20) (b, c); EK+W (d, e).
B. In the case of identical disk microresonators with equal parity of the field distribution (see fig. 3, a) relatively the plane: x - xs = 0 (s = 1,2), in the area: Az > L obtains
к
1,2
32i Pz
S1rwnml qz
(8)
•J
- n/i-K2k0 Az
0 /1 к2
{[(k2 - k2 )2 |C(к)|2 [J0(k0APK) ± (-1)m cOs(2mAV)J2m(k0 APK)] +
+
I D(k)|2 [J0 (k0Арк) + (- l)m cos(2mAv|/)J2m(k0Арк)] }кс!к
X
where
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Електродинаміка. ПристроїНВЧ діапазону. Антенна техніка
С( к) = [e0m
-Jm(p± )Jm(q ±к) - ho
1
1 mv-i- m v n у o /d2 i22\
P±koк ro(P - koк )
•[P±Jm(P±)jm(q1К) - q±Kjm'(P±)Jm(q±K)]} •
Pz sinp cosqzVl-к2 -k0Vl-к2 cospz sinq2
P2 -k2(1 -к2)
л/ї
к
Pz cosp sinqzл/l-к2 -k0Vl-к2 sinp cosqzVl—K
(9)
d(k) ^e^{[k0p±Jm(p± )jm(q ° - klq ±kJ^(p± )Jm(q 2)] •
l
P2 - k2(l-к2)
V>/l -K2 sin p cos qzVl -K2 - Pz cos p sin q Vl -K2 P sinp cosqzVl-к2 - k0л/l -к2 cosp sinqz Vl -к2
1
■ [p± Jm (p± )Jm(q ±K) - q A' (p± )Jm (q ±K)] •
P2 - (k0 K)2
k°Pz cos p sin qz Vl -к2 - k°k0Vl -к2 sin p cos qz Vl k°^s/l-к2 cospsinqz
к
:V1 -k2 -k2Pzsinpzcosq^/i-к
} -
-h0m
л/!-!2
(k2 - k2)
-Jm(p± )Jm(q 1 K) '
1
P2 - k2(l -к2)
p± ko K
Pz sin p cos qz Vl -к2 - k0 Vl -к" cos p sin qz k0Vl-k 2 sin p cos qz Vl -к2 - Pz cos p sin qz
2------- Vl -2
к
The integral convergence provides by choice of the radical signs for £ > 1:
-1 = —Ц/l-£2 in the (6, 7), as well as for Vk2 -1 = -iVl-к2 in the (8, 9). The coupling between disk microresonator and the open space can be ob-
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tained from:
к
32і pz
11
Sirwnml q
^ -J‘{(k2- k2)2 еь/Т^К2) + dьЯ^К2) }dK, (9)
following from Helmholtz-Kirchhoffs integral theorem.
Coupling coefficient analysis
Discovered relationships valid for any eigenoscillations of the disk microresonators in the open space, but a greatest interest presents a WG modes, as is well known, possessed a highest possible quality. It’s considering the coupling coefficients for the WG modes.
A = 0,2, for the HE^01 mode; a, b - for position, shown in fig. 2, a; c, d - for the position,
shown in fig. 3.
Fig. 2 - 4 are showing a coupling coefficient dependences on Disk microresonator centers, composed of the dielectric with the relative permittivity slr = 9,6 and the comparative dimensions A = L/2r0 = 0,2. It’s clear, that real part of the coupling coefficients has sufficiently visible values only in a small region, the resonator surfaces are closed with each other (fig. 2, b - e; fig. 3, b - c; fig. 4). Increasing distance between resonator centers accompanied by significant coupling coefficient decreasing. At that, relative motion in the tangent direction leads to complex interference (fig. 2, b, d; fig. 3, b; fig. 4, a, c) of their mutual influence, determining by significant eigenmode field variation nearby their sur-
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face. The greatest amounts of coupling between microresonators appears on its coaxial arrangement (fig. 3, b, c). The signs of the coupling coefficients extreme values determines both by the azimuth numbers and mutual microresonators position (fig. 2, c, e).
Imaginary part of the coupling coefficients are more smooth functions on coordinates (fig. 2, b, d; fig. 3, b; fig. 4, a, c). For selected dielectric permittivity its values approximately one tenth degrees as many as real parts. The same order have a coupling coefficient of the disk microresonator in the open space (9).
Filter parameters calculation
Obtained results allow to us create an electrodynamic models of various filters in the millimeter, terahertz or infrared wavelength ranges [13].
Fig. 5. Computed S-matrix parameters of the bandpass filter (a) on 15 Disk microresonators with EH^20l mode: slr = 9,6; QD = 109; A = 0,2; f0 = 200THz; kL = 10 6.
The fig. 5, 6 are showing results of the calculation of S-matrix bandpass filter parameters, that buildings up on a disk microresonators with EH (20 x mode.
Proposed, that coupling coefficients kL of the terminal microresonators with transmission lines are known. Mutual coupling coefficients was obtained from (6 - 9).
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Електродинаміка. ПристроїНВЧ діапазону. Антенна техніка
Fig. 6. Sketch of bandpass filter on vertically coupled disk microresonators (a). S-matrix responses of the 10-section bandpass filter with EHj+201 mode as functions of the wavelength (b - c). The coupling coefficients of the terminal resonators with transmission lines:
kL = 4 • 10-5; ^ = 9,6; QD = 109.
It's seen, that in consequence of rapidly coupling coefficients decreasing, all S-matrix parameters are symmetrical functions on the wavelength. As we used a large number of resonators, the s21 squareness was obtained very well.
Fig. 7. Computed S-matrix parameters of the bandstop filter (a) on 10 disk microresonators with EHj+29 j mode: slr = 9,6; QD = 1010; A = 0,2; f0 = 200 THz; kL = 106.
Fig. 7 shows a scattering parameters, calculated for the bandstop filter on 10 disk microresonators with EH+29 x mode. Obtained characteristics demonstrates
capability of the receiving narrowly band filters in the infrared range, that allows to use ones in the optical integrated circuits of various communication systems.
Conclusions
Analytical relationships for the coupling coefficients for WG modes of the disk microresonators in the Open space has been obtained and investigated.
It stated, that WG mode coupling coefficients describes by complicated de-
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pendencies on the structure parameters.
The real and imaginary parts of the coupling coefficients for the WG modes can be differed more than one tenth degrees.
The coupling between not adjacent microresonators in the filters is small in comparison with coupling between adjacent ones, that allows simply to build filters with symmetrically parameters of the scattering.
References
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2. Kirichenko A., Prokopenko Yu., Filipov Yu., Cherpak N. Qasioptical solid-state resonators, Kiev, Naukova dumka, 2008, 286 p.
3. Benson, T. M.; Boriskina, S. V.; Sewell, P.; Vukovic, A.; Greedy, S. C.; Nosich, A. I. Micro-Optical Resonators for Microlasers and Integrated Optoelectronics // Frontiers in Planar Lightwave Circuit Technology. - 2006. - 216. P. 39.
4. Smotrova E. I., Nosich A. I., Benson T. M., Sewell P. Optical Coupling of Whispering-Gallery Modes of Two Identical Microdisks and Its Effect on Photonic Molecule Lasing // IEEE J. of selected topics in Quantum Electronics. - 2006. - v. 12, N. 1. PP. 78 - 85.
5. Boriskina S.V. Coupling of whispering-gallery modes in size mismatched microdisk photonic molecules // Opt. Lett. - 2007. - 32. PP. 1557-1559.
6. Li J., Wang J., Huang Y. Mode coupling between first- and second-order whispering-gallery modes in coupled microdisks // Optics Letters. 2007. - June 1. Vol. 32, No. 11. PP. 1563-1565.
7. Preu S., Schwefel H. G. L., Malzer S., D'ohler G. H. and Wang L. J., Hanson M., Zimmerman J. D. and Gossard A. C. Coupled whispering gallery mode resonators in the Terahertz frequency range // Optics Express. - 2008. - 12 May. Vol. 16, No. 10. PP. 7336-7343.
8. Trubin A.A. Perturbation theory analysis of the System Coupling Dielectric Resonators // Bull. of National Technical University of Ukraine. Ser. Radiotechnique. - 1987. - V. 24. PP. 42-45.
9. Kajfez D., Guillon P.. Dielectric Resonators // Noble Publishing Corporation, Atlanta, 1998. 561 p.
10. Trubin A.A. Several specified integrals of special functions, http://eqworld.ipmnet.ru/ru/auxiliary/aux-specfunc.htm
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12. Ilchenko M.E., Trubin A.A., Electrodynamics' of Dielectric Resonators, Naukova Dumka, Kiev, 2004. 265 p.
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Трубін О. О. Коефіцієнти зв'язку дискових діелектричних резонаторів з коливаннями шепочучеї галереї. Приведено результати розрахунків коефіцієнтів взаємного зв'язку дискових діелектричних резонаторів з коливаннями шепочучей галереї. Розглянуті основні закономірності зміни зв'язку при варіації параметрів мікрорезона-торів. Розраховані частотні залежності матриці розсіювання смугового та ре-жекторного фільтрів на дискових микрорезонаторах інфрачервоного діапазону.
Ключові слова: мікрорезонатор, коефіцієнт зв'язку, коливання шепочучеї галереї,
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Електродинаміка. ПристроїНВЧ діапазону. Антенна техніка
S-матриця, фільтр
Трубин А. А. Коэффициенты связи дисковых диэлектрических резонаторов с колебаниями шепчущей галереи. Приведены результаты расчетов коэффициентов взаимной связи дисковых диэлектрических резонаторов с колебаниями шепчущей галереи. Рассмотрены основные закономерности изменения связи при вариации параметров микрорезонаторов. Рассчитаны частотные зависимости матрицы рассеяния полосовых и режекторного фильтров на дисковых микрорезонаторах инфракрасного диапазона.
Ключевые слова: микрорезонатор, коэффициент связи, колебания шепчущей галереи, S-матрица, фильтр
Trubin A. A. Coupling coefficients of the disk dielectric microresonators with whispering gallery modes. Calculation results of the Disk dielectric microresonators coupling coefficients are presented. Microresonator coupling coefficients as a functions of main parameters is considered. S-matrix frequency dependences of both bandstop and bandpass filters on Disk microresonators of infrared wavelength range is calculated.
Keywords: microresonator, coupling coefficient, whispering gallery mode, S-matrix, filter.
Вісник Національного технічного університету України «КПІ» Серія — Радіотехніка. Радіоапаратобудування. — 2015. — №61
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