Dielectric Resonator and Whispering Gallery Mode Resonator. This phenomenon of whispering gallery was first explained by Lord Rayleigh in

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Chapter 3 Dielectric Resonator and Whispering Gallery Mode Resonator Whispering Gallery is a acoustical feature associated with few monuments.

1 Chapter 3 Dielectric Resonator and Whispering Gallery Mode Resonator Whispering Gallery is a acoustical feature associated with few monuments. This phenomenon of whispering gallery was first explained by Lord Rayleigh in 1910, on the basis of his own observations made in an ancient gallery located under the dome of St. Paul s Cathedral in London (figure 3.1) that sound waves graze through concave surface of the wall and travel circumferentially along it [1]. Among the ancient Indian whispering galleries great Gol Gumbaz at Bijapur (figure 3.2) is most remarkable, and its architecture and acoustic features was explained by Sir C.V. Raman in 1922 [2]. (a) (b) Figure 3.1: (a) Image of whispering gallery at St. Paul s Cathedral in London, (b) sound path in circular whispering gallery, (source: 63

Acoustics of such enclosed places make them a whispering gallery where even a negligible sound like a whisper can be heard at diagonally opposite side of the Gumbaz or dome, moreover at the periphery

2 Acoustics of such enclosed places make them a whispering gallery where even a negligible sound like a whisper can be heard at diagonally opposite side of the Gumbaz or dome, moreover at the periphery of these dome are a circular balconies where any whisper, clap or sound gets echoed for around more than five times. For Gol Gumbaz, it is said that during the rule of sultan, Ibrahim Adil Shah, musicians used to sing, seated in the whispering gallery in order that the music produced could reach every corner of the hall. [1]- [3] Figure 3.2: Gol Gumbaz [3] In last few decades, studies based on Whispering Gallery modes (WGM) became popular due to its applications in optics and microwaves as ultrahigh Q-factor resonator. WGM resonators are also used for highly sensitive sensing applications by various researchers. These high-q sensors are usually used in identification of biomolecule and components of any material. Chapter 3 64

3 3.2 Dielectric Resonators Dielectric resonator (DR) is a kind of passive component to fix a certain frequency in the microwave systems. Dielectric resonator was first introduced by Richtmyer in 1939 [6]. He showed that dielectric materials of definite geometry can act as an electrical resonator at high frequencies. Richtmyer had also computed their resonant frequencies and losses for various uncomplicated structures and formulated a theory for dielectric resonators. In 1960 s, high frequency research was more focused around DR, with many researchers working on the development and applications of DR[7]. Okaya et.al showed that pieces of single crystals of rutile exibits high-q resonances at microwave frequencies and observed that decrease in the temperature will lead to increase in both the Quality factor (Q-factor) and the resonant wavelength. Following this many papers were reported by researchers like Jerzy Krupka, P. Guillon and Darko Kajfez [8] Chronological Review of the Work on WGM Dielectric Resonator Whispering gallery modes (WGM) are higher order modes found in DR and have similar characteristic as found in dome due to sound waves. Thus, modal fields confines within the small region near the boundary of DR. In 1967, J.R. Wait studied the electromagnetic WGM propagation in dielectric rod [9]. WGM in cylindrical dielectric resonators was first explained by J. Arnaud in 1982 [10]. After this many researcher had worked on this topic with various applications. This section gives a brief overview of the important work reported in the literature on dielectric WGM resonator. In 1987, Jiao et.al had discussed a theoretical method for calculating the resonant frequencies of WGM dielectric ring resonator [11]. In 1993, Eugene et.al had presented an analytical method for calculating the resonant frequency and Q-factor of Chapter 3 65

4 shielded dielectric disk resonator for higher order modes, at room and at cryogenic temperatures [12]. Then in 1994, Jerzy Krupka et.al had employed Rayleigh-Ritz method and finite element method capable of not generating spurious solutions for analysis of WGMs in cylindrical single crystal anisotropic dielectric resonator [13]. Imtiaz U Khairuddin et.al had explained a mathematical model of coupling between WGM-DR and a transmission line for application in millimetre wave monolithic integrated circuit (MMIC) [14]. In 1996, Darko Kajfez et.al had presented a comparison of analytical and numerical methods for higher order modes in dielectric resonator [15]. Tobar et.al had developed a composite DR with competing temperature dependent permittivity for possible applications in oscillators for reference[16]. They had successfully annulled the frequency temperature coefficient of a composite sapphire strontium titanate (Al 2 O 3 SrTiO 3 ) microwave resonator at 108 K with a resulting Q-factor of 20,000 50,000 below 150 K. In 2001, S.L Badnikar et.al had investigated resonant frequencies of dielectric disc utilizing the ring resonator model and results were used to generate a numerical expression for describing the operational frequencies for computer aided design applications [17] WGM mode in cylindrical DR Among theoretically investigated configurations, cylindrical shape has been commonly accepted as the most advantageous one. Therefore, in this section theory of WGM in cylindrical dielectric resonators, its basic properties and application of of higher order mode resonator is discussed. Realistic circuit applications often require mounting of resonator in the proximity of conducting walls or other dielectric materials. Thus, for accurate prediction of the resonant frequency it is essential to consider all these boundary perturbations in the theoretical analysis. Chapter 3 66

5 Dielectric constant of DR is usually taken very high as compared to air, due to which the reflection coefficient of the interface between DR body and air will go to unity thereby reflecting most of the energy inside the DR. Mathematically, as, 1 for DR reflection coefficient given by: = + = where, is characteristic impedance of air, Z is characteristic impedance of dielectric material and is relative permittivity of this material. Due to these reflections in DR, the standing waves are formed and EM resonance occurs. Resonant frequency of DR depends on following properties: i Resonance mode ii Size of DR iii Dielectric constant In WGM of cylindrical DR resonators, the modal field is confined within the small region near the resonator boundary. Ray optics, suggests that cause of this energy confinement is total internal reflection of a ray at the dielectric-air interface, and its movement is tangential to an inner caustic circle. Therefore, the ray moves only within a small region near the cylinder boundary, as shown in Fig From analytical approach, all waves guided in a dielectric cylinder can be described by Bessel functions ( ) where, argument will be of n order and for WGM case n >> 5. In WGM, modal field is oscillatory between the boundaries and a slightly smaller radius, while it decays exponentially elsewhere. Chapter 3 67

6 Modal ray O a Boundary a i Modal caustic Figure 3.3: WGM by ray optics Advantages of using WGM DR As stated by N.D Kataria et.al [18] in 2004: In WG mode, EM energy is confined around the dielectric-air interface inside the crystal because of total internal reflection that minimizes the radiation and conduction loss there by increasing the Q-value. With this understanding following are the broad advantages of WGM resonator: i. Good suppression of spurious modes. ii. Very high quality factor. iii. Sensitivity to the presence of absorbing and conducting materials etc. WGM are higher order modes of large azimuthal mode number n > 5 these modes are described as WGE nml and WGH nml mode. Where n represents azimuthal variations, m represents the radial variations and l represents axial variations. [9]-[11],[18] Chapter 3 68

7 3.3 Mathematical model of WGM cylindrical DR Mathematical model for the determination of resonant frequencies of the whispering-gallery modes of cylindrical dielectric resonator was proposed by Jiao, Guillon, and Bermudez [11], this model is known as JGB model. The advantage of this model is that it needs only real argument of the Bessel functions having integral order. Arrangement for solving JGB model is shown in figure 3.4. For the wave propagation in this circular cylinder, only regions 1, 2, and 3 are required however for cavity method region 4 should be considered due to truncation effects. Note that region 1 and 2 will be of same material in case of solid DR. z=h Region 4 ε r4 z=d z=0 z=-d Region 2 ε r2 Region 1 ε r1 Region 3 ε r3 ρ z=-h ρ= 0 ρ= a i Region 4 ε r4 ρ= a ρ= c Metallic wall Figure 3.4: Configuration of JGB model for Dielectric resonator. DR Model shown in Figure 3.4 have diameter 2a and height 2d, which is kept separated from wall of cavity and this cavity, is of diameter 2c and height 2h. In WGM-DR most of the energy is confined in region 1, between the cylindrical Chapter 3 69

8 dielectric-air boundary of radius a and the inner modal caustic of radius a i. Here, inner radius a i is an artificial "boundary" which divides the cylinder into the propagating and evanescent regions along radial direction. By solving the differential Bessel equation, the radius of the modal caustic can be estimated by = 3.2 where, n is the modal variation in the azimuthal direction, β the propagation constant and ε r the relative dielectric constant of the resonator. For the present study radial variation m and axial variation l are kept fixed whereas n is variable. Figure 3.5: (a) 3D view of Dielectric Resonator. (b) Components of fields for WGE and (b) WGH modes. Chapter 3 70

9 Figure 3.5 represents the 3D view of DR to get a better understanding of model given in figure 3.4 and also shows the direction of field s components for both WGE and WGH modes. Thus, by considering JGB model shown in figure 3.4 expressions for the electromagnetic field of WG modes can be establish. As usual, the longitudinal components of the field are obtained by solving the Helmholtz equation where, + =0 3.3 = In equation 3.3 can be either or. Let, the solution of wave equation is given by (,, )= ( ). ( ). ( ) hence, by using method separation of variable we will get three independent equations as solutions are given by ( )= + here, = +, we know that = for evanescent m odes and = for propagating mode. ( )= cos + sin ( )= (. )+ (. ) Chapter 3 71

10 here, = + whereas and represents the Bessel function of first kind and second kind respectively. If = then solution of ( ) will be given by ( )= (. )+ (. ) here, and are known as modified Bessel functions of first and second kind respectively. For WGE modes, the electric field is essentially transverse and E z may be neglected. Similarly, H z = 0 for WGH modes. Thus, for WGE n,m,l modes =0 3.4 =[ ( )+ ( )]sin cos 3.5 = ( )sin cos 3.6 = ( )sin cos 3.7 =[ ( )+ ( )]sin 3.8 and for WGH n,m,l modes =0 3.9 =[ ( )+ ( )]sin cos 3.10 = ( )sin cos 3.11 = ( )sin cos 3.12 Chapter 3 72

11 =[ ( )+ ( )]sin 3.13 with = = = = = = Transverse components can be deduced from E z and H z by = = 1 +1 = = 1 +1 Chapter 3 73

12 with = Boundary conditions for the calculation of resonant frequency, by matching the tangential components at = a and = a i, are as follows: = at = = = at = = =0 at = =0 for WGE n,m,l modes and = at = = = at = = =0 at = =0 for WGH n,m,l modes. Chapter 3 74

13 By solving these equations, following equation obtains: (. ) (. ) (. ) 0 (. ) (. ) (. ) 0 =0 (. ) (. ) 0 (. ) 3.14 (. ) (. ) 0 (. ) also (. )=0 and (. )=0 due to metallic cavity of radius c for WGE n,m,l modes. (. ) (. ) (. ) 0 (. ) (. ) (. ) 0 (. ) (. ) 0 (. ) (. ) (. ) 0 (. ) = From boundary conditions, fields become zero at metallic boundary for WGE n,m,l modes and for WGH n,m,l modes therefore (. )=0 and (. )=0 due to metallic cavity in determinant To obtain correct resonant frequency, we must take into consideration the axial energy confinement in region < d. From above field equation s longitudinal field components E z and H z respectively for TM and TE mode are proportional to G (z)= C cos in the region 1 and in region 4 it is G (z)= C here, C and C are proportionality constants. Now by applying following boundary conditions in axial direction by matching the longitudinal field components and their first derivatives at the discontinuity of the resonator radius: Chapter 3 75

14 ( )= ( ) ( ) = ( ) at z = d and ( )=0 at z = h ( ) =0 Thus, obtained equations are as follows: C cos = C C sin =C and, C =0 C ( ) =0 thus, following equations will obtained: tan = 3.16 and =0 Thus, from equations 3.5 and 3.10 it is observed that Bessel equation inside the dielectric resonator disc is oscillating for > and <, and from equations 3.6 Chapter 3 76

15 and 3.11 monotonically decreasing for < while in region 3 and 4 from fields equation 3.7, 3.8, 3.12 and 3.13 will decay exponentially. Final step is to solve and calculate the resonant frequency of WGM by solving simultaneously the equations 3.14 and 3.16 or 3.15 and 3.16 for the case of WGE and WGM mode correspondingly. Here, the values of c and h should be three times greater as comparison to dimensions of dielectric resonator according to the paper [12]. And this can also be verified by solving equations obtained from boundary conditions at the metallic cavity. 3.4 Application of WGM-DR s Dielectric resonators are widely used in oscillators, in filter designing and in complex permittivity measurement of dielectric materials. Due to these advantages of WGM-DR many researchers are using this technique in different applications some of which are as follows: In 1995, Taber and Flory had reported work on development of commerciallyviable high purity X-band signal source, by integrating a WGM cryogenic sapphire dielectric resonator. WGM are used due to its tremendous electromagnetic field confinement capability and using this they had developed microwave oscillators.[19] WGM DR are also used in oscillator applications, In 1998, Tobar et.al had reported a method to determine the dielectric properties of a single crystal rutile (TiO2) resonator using whispering gallery modes [20]. This work gave impetus to research on determining dielectric properties of various types of materials and fluids: In 1999, Krupka et.al had reported a work in which whispering gallery modes were Chapter 3 77

16 used for very accurate measurement of permittivity and dielectric loss of ultralow loss isotropic and uniaxially anisotropic single crystals at cryogenic temperatures [21]. Again in 1999 Krupka et.al had published similar work in which they had obtained the relationship between resonant frequencies and calculated the permittivity of the sample under test with a radial mode-matching technique [22] and Ratheesh et.al had published a paper on the microwave dielectric properties of Ba(Mg 1/3, Ta 2/3) O 3 (BMT) ceramic resonators by utilizing the WGM and conventional methods in the frequency range 6 18 GHz [23]. Then in year 2000, Annino et.al had characterized dielectric properties of materials using whispering gallery dielectric resonators and the field distribution in the different WG resonant modes was obtained by an analytical calculation under the mode matching method approximation [24]. Prokopenko et.al had presented their paper on WGM dielectric resonators for the millimetre wave near field sensing applications in According to them resonant parameters of both the isotropic and anisotropic DR depends on the resonant frequency, Q-factor, slow-down factor and electric energy filling factor of the modes [25]. Then in 2003, Krupka had discussed on various resonant measurement methods for complex permittivity determination of lossy dielectrics [26]. Then in 2006, his review article was published on microwave frequency domain measurement techniques of the complex permittivity, in which he had mentioned all various methods including WGM method [27]. In 2008, Shaforost et.al had presented a paper on fingerprint detection and analysis of biochemical (lossy) liquids of pico-to-nanoliter volumes using open whispering-gallery-modes resonator with local inhomogeneities by using a liquid Chapter 3 78

17 droplet. According to authors, this liquid droplet causes a small perturbation of the electromagnetic field distribution which will lead to change in Q-factor and resonant frequency of resonator [28]. Then they had published similar work in October 2008 [29]. After this in 2009, Shaforost et.al had presented a paper on WGM resonators for evanescent sensing of nanolitre liquid substances [30] and published a paper on similar work in which a microwave resonator composed of a sapphire cylinder and a quartz plate with a 400 nano litre cavity was introduced for the determination of the complex permittivity of liquids at 10 GHz [31]. In 2010, Mohamed S. Kheir et.al. presented a paper in which a WGM dielectric resonator was proposed as a biological material sensor, for this they had created a 2mm*2mm cavity in dielectric resonator for calculating the permittivity value [32]. In 2010, another paper was published on WGM resonance sensor for dielectric sensing of drug tablets, by Mohammad Neshat et.al. In this they had used two methods for measuring the properties of drug tablet by putting it on top of a dielectric disk resonator and inside a dielectric ring resonator, on the basis of resonance frequency and Q-factor of the composite sample tablet and resonator arrangement [33]. Chapter 3 79

18 References [1] C. V. Raman and G. A. Sutherland, On the Whispering Galleries Phenomenon, Proceedings of Royal Society London A, vol. 100, pp , [2] C.V Raman, On Whispering Galleries, Bulletin of Indian Association for the Cultivation of Science, vol. 7, pp , [3] Gol Gumbaz, [Online] Available: [Nov 11, 2011]. [4] A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Review of Applications of Whispering-Gallery Mode Resonators in Photonics and Nonlinear Optics, IPN Progress Report , August 15, [5] Frank Vollmer and Stephen Arnold, Whispering-Gallery-Mode Biosensing Labelfree Detection Down To Single Molecules, Nature Methods, vol.5, no.7, July [6] R.D. Richtmyer, Dielectric Resonators, Journal of Applied Physics, vol. 10, pp , June [7] A. Okaya and L.F. Barash, The Dielectric Microwave Resonator, Proceedings of the IRE, vol. 50, pp , Oct [8] Darko Kajfez and Piere Guillon, Dielectric Resonators, 2 nd ed. Georgia, USA: Noble, [9] J.R. Wait, 'Electromagnetic whispering-gallery modes in a dielectric rod', Radio Science, vol.2, pp , [10] C. Vedrenne, and J. Arnaud, 'Whispering-gallery modes of dielectric resonators', IEE Proc. H Microwaves, Antennas and propagation, vol.129, no. 4, pp , Chapter 3 80

19 [11] X.H. Jiao, P. Guillon, L.A. Bermudez, Resonant frequencies of whisperinggallery dielectric resonator modes, IEE Proceedings, vol. 134, Pt. H, no. 6, pp , Dec [12] Eugene N. Ivanov, David G. Blair, and Victor I. Kalinichev, Approximate Approach to the Design of Shielded Dielectric Disk Resonators with Whispering-Gallery Modes, IEEE Transactions on Microwave Theory And Techniques, vol. 41, no. 4, pp , April [13] Jerzy Krupka, Dominique Cros, Michel Aubourg, and Pierre Guillon, Study of Whispering Gallery Modes in Anisotropic Single-Crystal Dielectric Resonators, IEEE Transactions On Microwave Theory And Techniques, vol. 42, no. 1, pp , Jan [14] Imtiaz U Khairuddin, Ian C Hunter, A Theoretical Model for Dielectric Resonators in Whispering Gallery Mode for Application in Millimetre Wave Monolithic Integrated Circuits, in IEE Colloquium on Modelling, Design and Application of MMIC's, 1994, pp. 7/1-7/7. [15] Darko Kajfez, Atef Elsherbeni, and Asem Mokaddem, Higher Order Modes in Dielectric Resonators, in Antennas and Propagation Society International Symposium, AP-S. Digest,1996, vol.1, pp [16] M E Tobar, J Krupka, E N Ivanov and R A Woode, Dielectric frequency temperature compensated microwave whispering-gallery-mode resonators, Journal of Physics D: Applied Physics, vol. 30, pp , [17] S.L. Badnikar and N. Shanmugam and V.R.K. Murthy, Resonant frequencies of whispering-gallery modes dielectric resonator, Defense Science Journal, vol. 51, no.2, pp , April [18] N.D Kataria and Vijay Kumar, Frequency Temperature Compensated Whispering Gallery Mode Dielectric Resonator Oscillator, Chinese Journal of Physics, vol. 42, no. 4-II, pp , Aug [19] R. C. Taber and C. A. Flory, Microwave Oscillators Incorporating Cryogenic Sapphire Dielectric Resonators, IEEE Transactions On Ultrasonics, Ferroelectrics, And Frequency Control, vol. 42, no. 1, pp , Jan [20] Michael Edmund Tobar, Jerzy Krupka, Eugene Nicolay Ivanov, and Richard Alex Woode, Anisotropic complex permittivity measurements of mono- Chapter 3 81

20 crystalline rutile between 10 and 300 K, Journal of Applied Physics, vol. 83, no. 3, pp , Feb [21] Jerzy Krupka, Krzysztof Derzakowski, Michael Tobar, John Hartnett and Richard G Geyer, Complex permittivity of some ultralow loss dielectric crystals at cryogenic temperatures, Measurement Science and Technology, vol. 10, no. 5, pp , [22] Jerzy Krupka, Krzysztof Derzakowski, Adam Abramowicz, Michael Edmund Tobar, and Richard G. Geyer, Use of Whispering-Gallery Modes for Complex Permittivity Determinations of Ultra-Low-Loss Dielectric Materials, IEEE Transactions On Microwave Theory And Techniques, vol. 47, no. 6, pp , June [23] R Ratheesh, M T Sebastian, M E Tobar, J Hartnett and D G Blair, Whispering Gallery mode microwave characterization of Ba(Mg1=3,Ta2=3)O 3 dielectric resonators, Journal of Physics D: Applied Physics, vol. 32, no. 21, pp , [24] G. Annino, D. Bertolini, M. Cassettari, M. Fittipaldi, I. Longo, and M. Martinelli, Dielectric properties of materials using whispering gallery dielectric resonators: Experiments and perspectives of ultra-wideband characterization, Journal of Chemical Physics, vol. 112, no. 5, pp , Feb [25] Yu. Prokopenko, M. F. Aka, S. N. Kharkovsky, Whispering Gallery Mode Dielectric Resonators For The Millimeter Wave Near Field Sensing Apllications, in The Fourth International Kharkov Symposium on Physics and Engineering of Millimeter and Sub-Millimeter Waves, 2001, vol. 2, pp [26] Jerzy Krupka, Precise measurements of the complex permittivity of dielectric materials at microwave frequencies, Materials Chemistry and Physics, vol. 79, no.1-2, pp , [27] Jerzy Krupka, Frequency domain complex permittivity measurements at microwave frequencies, Measurement Science and Technology, vol. 17, no. 6, pp. R55 R70, [28] Elena N. Shaforost, Alexander A. Barannik, Svetlana Vitusevich, Andreas Offenhäusser, Open WGM Dielectric Resonator Technique for Characterization of nl-volume Liquids, in 38th European Microwave Conference, 2008, pp Chapter 3 82

21 [29] E. N. Shaforost, N. Klein, S. A. Vitusevich, A. Offenhäusser, and A. A. Barannik, Nanoliter Liquid Characterization by Open Whispering-Gallery Mode Dielectric Resonators at Millimeter Wave Frequencies, Journal Of Applied Physics, vol. 104, no.7, pp , [30] Elena N. Shaforost, Norbert Klein, Alexey I. Gubin, Alexander A. Barannik, Alexander M. Klushin, Microwave-Millimetre Wave WGM Resonators for Evanescent Sensing of Nanolitre Liquid Substances, in 39th European Microwave Conference, 2009, pp [31] E. N. Shaforost, N. Klein, S. A. Vitusevich, A. A. Barannik, and N. T. Cherpak, High sensitivity microwave characterization of organic molecule solutions of nanoliter volume, Applied Physics Letters, vol. 94, no. 11, pp , [32] Mohamed S. Kheir, Hany F. Hammad, and Abbas Omar, Experimental Investigation of Whispering-Gallery-Mode Dielectric Resonators for Biological Material Characterization, in Conference on Precision Electromagnetic Measurements, 2010, pp [33] Mohammad Neshat, Huanyu Chen, Suren Gigoyan, Daryoosh Saeedkia and Safieddin Safavi-Naeini, Whispering-gallery-mode resonance sensor for dielectric sensing of drug tablets, Measurement Science and Technology, vol. 21, no. 1, pp , Chapter 3 83

Temperature Coefficient Measurement of Microwave Dielectric Materials Using Closed Cavity Method

Progress In Electromagnetics Research Letters, Vol. 63, 93 97, 2016 Temperature Coefficient Measurement of Microwave Dielectric Materials Using Closed Cavity Method Liangzu Cao 1, 2, *, Junmei Yan 1, and