Resonant frequencies of whispering-gallery dielectric resonator modes X.H. Jiao P. Guillon L.A. Bermudez Indexing terms: Millimetre wave devices and components, Microwave integrated circuits, Dielectric materials Abstract: Whispering-gallery dielectric resonator modes have shown their usefulness in millimetre wave integrated circuits. In the paper, a method for the determination of the resonant frequencies of the whispering-gallery modes of cylindrical dielectric resonators is presented. Theoretical and experimental results are given. A comparison with other methods is also made. However, it fails to predict some resonances at higher fre- quencies, in particular, for resonators of small diameter. 1 Introduction Cylindrical dielectric resonators made of high permit- tivity (e r = 36) dielectric materials have been developed in recent years, in particular low loss and temperature stable dielectric materials [1]. They are used in many passive and active microwave components. The advan- tages offered by such dielectric resonators, such as small size, temperature compensation, good field concentration, high quality factor and easy integration make them very useful in microwave integrated circuits. However, the use of such conventional resonators is significantly reduced at millimetre wavelengths, because, if the cylindrical resonators are to continue to act on TE, TM or hybrid modes, their dimensions must be impracti- cally small. Whispering-gallery (WG) modes dielectric resonators, proposed by J. Arnaud et al. [5-6], are very suitable for millimetre wave integrated circuits, overcoming the most serious defect of conventional resonators in this fre- quency band. In fact, dielectric resonators operating on whispering-gallery modes have a geometry different from the conventional resonators, as shown in Fig. 1, and their dimensions are 'oversize' for millimetre wavelengths. Other advantages offered by the WG Modes, such as good suppression of spurious modes, very high quality factor and insensitivity to the presence of absorbing and conducting materials make such resonators very inter- esting for millimetre wave integrated circuits. In Reference 5, Arnaud et al. proposed a method for the determination of the resonant frequencies of the WG dielectric resonator modes. This method, based on solving the eigenvalue equation of a circular dielectric rod for different propagation constants, gives accurate resonant frequencies for resonators of large diameter. Paper 5686H (El2), first received 30th January and in revised form 13th July 1987 The authors are with LCOM-CNRS UA356 UER des sciences, Uni- versite de Limoges, 123 Av. Albert Thomas, 87060 Limoges, Cedex, France Fig. 1 Geometry of a Whispering-gallery mode cylindrical dielectric resonator In this paper, by establishing a model for whispering- gallery mode dielectric resonators, we present a method to compute the resonant frequencies of the WG modes of cylindrical dielectric resonators. Applications to several resonators of different sizes are discussed and theoretical and experimental results given for comparison. 2 Whispering-gallery modes of dielectric resonators Whispering-gallery modes in acoustics were discovered by Lord Rayleigh in 1910. He found that high frequency sound waves have a tendency to cling to a concave surface [2]. In 1967, J.R. Wait studied the analogous pro- pagation in dielectric cylinders [4]. It was J. Arnaud who began studies on whispering-gallery modes in cylindrical dielectric resonators in 1981 [5]. The whispering-gallery (WG) modes of cylindrical dielectric resonators comprise waves running around the concave side of the cylindrical boundary of the reson- ators. The waves essentially move in the plane of the cir- cular cross-section. Most of the modal energy remains confined between the cylindrical boundary and a modal caustic. In other words, the modal field is essentially kept concentrated within a small region near the resonator boundary. IEE PROCEEDINGS, Vol. 134, Pt. H, No. 6, DECEMBER 1987 497
Transcript
Resonant frequencies of whispering-gallerydielectric resonator modes
Abstract: Whispering-gallery dielectric resonatormodes have shown their usefulness in millimetrewave integrated circuits. In the paper, a methodfor the determination of the resonant frequenciesof the whispering-gallery modes of cylindricaldielectric resonators is presented. Theoretical andexperimental results are given. A comparison withother methods is also made.
However, it fails to predict some resonances at higher fre-quencies, in particular, for resonators of small diameter.
1 Introduction
Cylindrical dielectric resonators made of high permit-tivity (er = 36) dielectric materials have been developed inrecent years, in particular low loss and temperaturestable dielectric materials [1]. They are used in manypassive and active microwave components. The advan-tages offered by such dielectric resonators, such as smallsize, temperature compensation, good field concentration,high quality factor and easy integration make them veryuseful in microwave integrated circuits.
However, the use of such conventional resonators issignificantly reduced at millimetre wavelengths, because,if the cylindrical resonators are to continue to act on TE,TM or hybrid modes, their dimensions must be impracti-cally small.
Whispering-gallery (WG) modes dielectric resonators,proposed by J. Arnaud et al. [5-6], are very suitable formillimetre wave integrated circuits, overcoming the mostserious defect of conventional resonators in this fre-quency band. In fact, dielectric resonators operating onwhispering-gallery modes have a geometry different fromthe conventional resonators, as shown in Fig. 1, and theirdimensions are 'oversize' for millimetre wavelengths.Other advantages offered by the WG Modes, such asgood suppression of spurious modes, very high qualityfactor and insensitivity to the presence of absorbing andconducting materials make such resonators very inter-esting for millimetre wave integrated circuits.
In Reference 5, Arnaud et al. proposed a method forthe determination of the resonant frequencies of the WGdielectric resonator modes. This method, based onsolving the eigenvalue equation of a circular dielectricrod for different propagation constants, gives accurateresonant frequencies for resonators of large diameter.
Paper 5686H (El2), first received 30th January and in revised form 13thJuly 1987
The authors are with LCOM-CNRS UA356 UER des sciences, Uni-versite de Limoges, 123 Av. Albert Thomas, 87060 Limoges, Cedex,France
Fig. 1 Geometry of a Whispering-gallery mode cylindrical dielectricresonator
In this paper, by establishing a model for whispering-gallery mode dielectric resonators, we present a methodto compute the resonant frequencies of the WG modes ofcylindrical dielectric resonators. Applications to severalresonators of different sizes are discussed and theoreticaland experimental results given for comparison.
2 Whispering-gallery modes of dielectricresonators
Whispering-gallery modes in acoustics were discoveredby Lord Rayleigh in 1910. He found that high frequencysound waves have a tendency to cling to a concavesurface [2]. In 1967, J.R. Wait studied the analogous pro-pagation in dielectric cylinders [4]. It was J. Arnaud whobegan studies on whispering-gallery modes in cylindricaldielectric resonators in 1981 [5].
The whispering-gallery (WG) modes of cylindricaldielectric resonators comprise waves running around theconcave side of the cylindrical boundary of the reson-ators. The waves essentially move in the plane of the cir-cular cross-section. Most of the modal energy remainsconfined between the cylindrical boundary and a modalcaustic. In other words, the modal field is essentially keptconcentrated within a small region near the resonatorboundary.
IEE PROCEEDINGS, Vol. 134, Pt. H, No. 6, DECEMBER 1987 497
This energy confinement can be explained from a ray-optics point of view [5]. A ray is totally reflected at thedielectric-air interface, and it is then tangent to an innercircle called a caustic. Thus, the ray only moves within asmall region near the rod boundary, as shown in Fig. 2.
modal ray
boundary
modal caustic
Fig. 2 Whispering-gallery modes by ray optics
The confinement may also be explained from an ana-lytical point of view. As we know, all waves guided in adielectric rod can be described by Bessel functions Jn(kr).The WG modes correspond to those for which the argu-ment kr is of the order of n, which is a large integer. Inthis case, the Bessel functions can correctly be approx-imated by Airy functions. In this manner, it has beenshown that the modal field is oscillatory between theboundary and a slightly smaller radius, while it decaysexponentially elsewhere [3].
The WG dielectric resonator modes are also charac-terised by energy confinement within a small region inthe axial direction of the resonator. This axial confine-ment is achieved by increasing the diameter of the dielec-tric rod slightly in some central region \z\ < d. In thisconfiguration, the resonant mode propagates with asmall propagation constant along the rod axis within thisenlarged region while it decays exponentially outside thisregion.
Apart from 'oversize' dimensions, dielectric resonatorsoperating on WG modes also offer good suppression ofspurious modes which, leaking out of the resonator, canbe absorbed without disturbing the desired modes. TheWG modes have a very high quality factor which islimited only by the intrinsic loss, because the radiationlosses are almost negligible. Besides, the modes are notsensitive to the presence of conducting and absorbingmaterials, owing to the confinement of modal energywithin the resonator.
The WG modes of dielectric resonators are classed aseither WGE n m / > ± 1 or WGHn m , ± 1 . For WGE modes,the electric field is essentially transverse while for WGHmodes, the electric field is essentially axial. Here theinteger n denotes the azimuthal variations of modes, mthe radial variations and / the axial ones. Finally, the twopossible rotating senses of these modes are denoted by± 1 . However, this designation will be omitted, becausein an isotropic dielectric medium the resonant frequenciesof the modes are the same whatever the sense of rotation.In the millimetre wave frequency band, these WG modescan be excited by synchronising with an external trav-elling wave source by means of microstrip lines, meanderlines, dielectric image guides or other transmission lines.
3 Resonant frequencies of whispering-gallerymodes
In Reference 5, resonant frequencies of the WG modes ofdielectric resonators were investigated. The method con-sists in solving the eigenvalue equation of a uniformdielectric cylinder for different propagation constants andaccounting for the central cylinder enlargement bymatching the field and its first derivative at the discontin-uity of the diameter. However, this method fails to giveall resonances for resonators of small diameter.
The proposed method involves taking the resonatorunder consideration as a ring stucture resonator. Theexternal diameter of the ring coincides with the diameterof the real resonator, while the internal one takes thevalue of the caustic of the mode being considered. Bymaking this idealisation and writing out the field expres-sions for this dielectric ring structure, we are able toobtain the equations which give the resonant frequencyfor a particular mode.
3.1 A model of the WG mode dielectric resonatorAs mentioned above, WG modes move essentially in theplane of a circular cross-section, and most of the modalenergy is confined between the resonator boundary andthe inner modal caustic. As a result, the WG mode dielec-tric resonator is analogous to a travelling wave resonatorwith a dielectric ring structure, an analogy which is justi-fied by the energy confinement, the existence of themodal caustic and the fact that the field of the WG modedecays exponentially outside the enlarged region [7]. Forthis dielectric ring, the external diameter coincides withthe diameter of the real dielectric resonator, while theinternal one takes the place of the modal caustic, asshown in Fig. 3.
Fig. 3model
a b
Whispering-gallery mode dielectric resonator and the idealised
a Real resonatorb Resonator model
Region \:\z\<d,a,<r <aRegion 2: \z\<d,r<al
Region 3 : | z | < d, r > aRegion 4: | z | > <f, at<r < a
By using a ray optics technique or by solving the dif-ferential Bessel equation, the radius of the modal causticcan be estimated by
a, = (1)
498 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 6, DECEMBER 1987
where n is the modal variation in the azimuthal direction,P the propagation constant and er the relative dielectricconstant of the resonator.
This analogy can be considered as a model of dielec-tric resonators operating in WG modes. It is the basis ofall our studies concerning whispering-gallery modesincluding those of resonant frequencies and coupling withtransmission lines. This model can also be used to helpunderstand the operating principles of niters making useof WG mode dielectric resonators.
3.2 Electromagnetic field of WG modesWe consider the ring model of Fig. 3b and establishexpressions for the electromagnetic field of WG modesfor this model. As usual, the longitudinal components ofthe field are obtained by solving the Helmholtz equation
with
For WG modes it is important to take into account theirpropagation characteristics: in cross-section, the field ofWG modes is oscillatory between the caustic and the res-onator boundary, and evanescent elsewhere, while alongthe rod axis, the resonant mode propagates only withinthe enlarged region, and decays exponentially on bothsides of this region. In addition, for WGE modes, theelectric field is essentially transverse and Ez may beneglected. Similarly we have Hz = 0 for WGH modes.Thus we have [8] for WGEfli m_, modes
Ez = 0
Hzl = \_AeJn(k,r) + Be r n (V)] exp {jnd) cos {fiz)
Hz2 = Ce In(k2 r) exp (jnd) cos {fiz) (2a)
Hz3 = De Kn(k3 r) exp {jnO) cos {fiz)
tf24 = MUM + B'e YJLks)} exp (jnd) exp (-<xz)
and for WGHn m , modes
c
' - lk2F -*1 — V 0 crl
k2 =
- k2 (2c)
a. — Ik2 — k2
The transverse components can be deduced from Ez andHzby
I 1 dH2 d2Ez
8HZ ld2Ez
1 dEz
dE.
82HZ1
Jd2Hz
with
3.3 Resonant frequenciesNow it is time to look for the equations which give theresonant frequency for a given mode. For this, we matchthe tangential components at r = a and r = a,-, that is
respectively for WGEn m , modes and for WGHn m ,modes.
In order to obtain the correct resonant frequency, wemust take into consideration the axial energy confine-ment in region \z\ < d. As did Arnaud, we match the fieldand its first derivative at the discontinuity of the reson-ator radius. Keeping in mind that the essential field com-ponents for WGE modes are Hz, Er and E^,, and thosefor WGH modes are E2,Hr, H^ we have
<Di(z) = <D4(z)
dz dz
with
O^z) = A cos (jSz)
<D4(z) = B exp ( — CLZ)
Thus, we obtain
a(5)
for both WGEn m , modes and WGHn m , modes.Thus, we are able to find the resonant frequencies of
the whispering-gallery modes of cylindrical dielectric res-onators by simultaneously solving eqns. 3 and 5 or 4 and5.
4 Results and comparison
The theory was applied to three resonators of differentsizes. Theoretical and experimental results are givenrespectively in Tables 1, 2 and 3. In these tables, theoreti-
Table 1a: Resonant frequencies of WGE. „ ,
Present method Measured Method in Reference 5
Modes F, GHz F, GHz F, GHz Modes
WGE3,0,0WGE4,0,0
WGE5,0,0
WGE6.0.0
WGE7,0 j 0
WGE8.0 i 0
WGEQ „ „
3.754.415.045.656.256.857.49
3.714.375.025.636.256.827.44
3.624.254.875.476.056.637.21
WGE3i0>0WGE4.0,0
WGE5>OiO
WGE6,0>0
WGE7,OiO
WGE8,0.0
WGEn „ „
Er = 38, a = 15 mm, a, = 13 mm, d = 4 mm
Table 1b: Resonant frequencies of W H G , . ,
Present method Measured Method in Reference 5
Modes F, GHz F, GHz F, GHz Modes
WGH, 0 0WGH2;0 0
WGH 3 0 0WGH 4 0 0W G H 5 ; O I O
WGH 6 i 0 , 0
W G H 7 0 0
3.143.794.415.025.626.216.80
3.063.774.455.105.746.356.96
2.853.544.194.795.406.006.59
WGH3.0 , 0WGH4 0 0
W G H 5 > 0 0
W G H 6 0 ' 0
WGH 7 I 0 ; 0
WGH8 0 0
W G H 9 0 0
Er = 38, a = 15 mm, a, = 13 mm, d = 4 mm
500
= 0 (4)
Table 2a: Resonant frequencies of WGE, „
Present method
Modes
WGE5.0,0
WGE6,0 i0
WGE7i0>0
WGE8>0>0
Er = 36, a =
Table 2b:
F, GHz
8.3559.381
10.38811.380
Measured
F, GHz
8.1019.125
10.11211.196
, i
Method in Reference 5
F, GHz
8.2739.346
10.40211.410
= 9.21 mm, ay = 8.21 mm, d = 2.7 mm
Resonant frequencies of W G H . ,
Present method
Modes F,GHz
Measured
F, GHz
Method in
F, GHz
Modes
WGE9,0 i 0
WGE6,0>0
WGE7>0>0
WGE8 i 0 j 0
» , i
Reference 5
Modes
W G H 4 i 0 0
WGH 5 > 0 0
WGH6 0 0
WGH 7 > 0 0
WGH8 0 0
8.2949.301
10.29511.28012.257
8.2459.285
10.28511.25312.244
8.4399.547
10.62911.70612.763
WGH6 0 0
W G E 7 0 0
W G H 8 0 0
W G H 9 0 0
WGH, 0 i 0 > 0
Er = 36, a = 9.21 mm, a, = 8.21 mm, cr = 2.7 mm
Table 3: Resonant frequencies of WGE. „ ,
Present method Measured Method in Reference 5
Modes F, GHz F, GHz F, GHz Modes
W G E 4 0 i 0
WGE2 1 0
WGE5tOiO
WGE3 , 0
W G E 6 ; O I O
W G E 7 0 0
26.52228.60730.27132.84633.90837.466
26.76628.65230.84433.07334.73037.300
•
28.26031.804
*35.30138.722
*WGE5,0,0
WGE6,0>0
*WGE7,0,0
W G E 8 0 0
Er = 36, a = 2.63 mm, a, = 2.30 mm, d = 0.7 mm
cal resonant frequencies calculated from the method ofReference 5 are also presented for comparison.
In Table 1, the measured resonant frequencies wereobtained by Arnaud et ah, and excitation and detectionwere by means of electric and magnetic probes [5]. InTable 2, the dielectric resonators were excited anddetected by microstrip line. In the millimetre wave fre-quency band, dielectric image guide was used as trans-mission line. Measured results are given in Table 3.
From the above tables, it is seen that the theoreticalresults obtained from the present theory are in goodagreement with the experimental results.
5 Conclusion
A method determining the resonant frequencies of thewhispering-gallery modes of cylindrical dielectric reson-ators was presented. Theoretical and experimental resultswere shown to be in good agreement.
In the millimetre wave frequency band, thewhispering-gallery dielectric resonator modes can beexcited by microstrip lines, dielectric image guides orother millimetre wave transmission lines. Such dielectricresonators can be used in high performances millimetrewave components, such as bandstop filters, directionalfilters and power combiners.
IEE PROCEEDINGS, Vol. 134, Pt. H, No. 6, DECEMBER 1987
6 References
1 KAJFEZ, D., and GUILLON, P.: 'Dielectric resonators' (ArtechHouse, 1986)
2 Lord RAYLEIGH: 'The problem of the Whispering Gallery', Philos.Mag., 1910, 20, pp. 1001-1004
3 Lord RAYLEIGH: 'Further applications of Bessel functions ofhigher order to the Whispering Gallery and allied problems', Philos.Mag., 1914, 27, pp. 100-104
4 WAIT, J.R.: 'Electromagnetic whispering-gallery modes in a dielectricrod', Radio Sci., 1967, 2, pp. 1005-1017
5 VEDRENNE, C , and ARNAUD, J.: 'Whispering-gallery modes ofdielectric resonators', IEE Proc. H Microwaves, Antennas and propa-gation, 1982,129, (4), pp. 183-187
6 ARNAUD, J.: 'Beam and fiber optics' (Acandemic Press, NY, 1976)7 JIAO, X.H., GUILLON, P , and OBREGON, J.: 'Theoretical
analysis of the coupling between whispering-gallery dielectric reson-ator modes and transmission lines', Electron. Lett., 1985, 21, (3), pp.88-89
8 HARRINGTON, R.F.: 'Time-harmonic electromagnetic fields'(McGraw Hill, NY, 1961)
Errata§EKER, S., and SCHNEIDER, A.: 'Stochastic model forpulsed radio transmission through stratified forests', IEEProc. H, Microwaves, Antennas & Propag., 1987, 134, (4),pp. 361-368
In the integral of eqn. 6 the numerator should be multi-plied by a factor of [1 + Rc Ra + (Rc + £fl)]. Eqns. 30and 31 should begin with 60/ and 120/, respectively
5486H
DAMIANO, J.P.: 'Computation of input impedance inmicrostrip antennas. Graphic representation and numeri-cal integration of oscillating functions', IEE Proc. H,Microwaves, Antennas & Propag., 1987, 134, (5), pp. 456-466
Eqns. 18a and 186 should read
*(*„ ky) = \Fx(kx, ky)\2 cos (kx(xr - xj)
x cos (ky(yr - yj)
Too To
J>/(8r) JJ:,ky)R(kx,ky)dkxdky
Lexx\kx> ky) e)
x R(kx, ky)dkxdky
J'oo f oo
efxYMP(K,ky)
v/(«r) JWer)
x R(kx,ky)dkxdk
xx '(*„
The multiplication sign in eqn. 26 should be replaced byan equals sign and the third integral in 4.3(i) should be
5546H
(18a)
IEE PROCEEDINGS, Vol. 134, Pt. H, No. 6, DECEMBER 1987 501
