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Normal Modes and Quality Factors of Spherical Dielectric Resonators: I – Shielded Dielectric Sphere

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Normal Modes and Quality Factors of Spherical Dielectric Resonators: I – Shielded Dielectric Sphere PRAMANA °c Indian Academy of Sciences Vol. 62, No. 6 | journal of June 2004 physics pp. 1255{1271 Normal modes and quality factors of spherical dielectric resonators: I { Shielded dielectric sphere R A YADAV and I D SINGH Spectroscopy Laboratory, Department of Physics, Banaras Hindu University, Varanasi 221 005, India E-mail: [email protected] MS received 16 July 2002; revised 27 October 2003; accepted 1 April 2004 Abstract. Electromagnetic theoretic analysis of shielded homogeneous and isotropic di- electric spheres has been made. Characteristic equations for the TE and TM modes have been derived. Dielectric spheres of radii of the order of ¹m size are found suitable for the optical frequency region whereas for the microwave region radii of the order of mm size are found suitable. Parameters suitable for their application in the optical and microwave frequency ranges have been used to compute the frequencies corresponding to the normal modes for the TE and TM modes. Expressions for the quality factors for realistic res- onators, i.e., for a dielectric sphere with a non-zero conductivity and a metal shield with a ¯nite conductivity have also been derived for the TE and TM modes. Computations of the quality factors have been made for resonators with parameters suitable for the optical and the microwave regions. Keywords. Eigenmodes; spherical resonators; spherical dielectric resonators; quality factors. PACS No. 42.50.Dv 1. Introduction The earliest reported study on spherical resonators seems to be that of Debye [1] wherein he has studied normal modes of a conducting sphere embedded in a perfect dielectric medium. Stratton has treated the above case and the case of oscillations of a spherical cavity in his classic text [2]. While discussing the cavity resonators, Waldron [3] has considered the case of the spherical homogeneous simple perfect cavity and has derived the ¯eld expressions and the characteristic equations for such a cavity based on the treatment of Bromwich [4]. A dielectric sphere with a given dielectric constant and radius possesses natural modes of oscillation having charac- teristic frequencies. Such oscillations are known as structure resonances and these have been studied both theoretically and experimentally in the microwave region [5,6] and more recently in the optical region of the electromagnetic spectrum [7{10]. 1255 R A Yadav and I D Singh Structure resonances have been studied using fluorescence [11], optical levitation [12], absorption [13] and scattering [14]. Electromagnetic ¯eld analysis of spherical dielectric resonators has been pre- sented by a number of workers [15]. In all the above cases the resonant frequencies and the quality factors have been computed for the resonators with parameters suitable for the microwave region. From the survey of the published literature it seems that no theoretical and/or experimental studies are available for the optical frequency region. In addition neither the normal mode frequencies nor the quality factors in the optical region seem to have been reported for shielded spherical di- electric resonators. Shielded resonators have drastically reduced quality factors due to metallic loss of the shield and dielectric loss of the dielectric medium. However, cavity resonators with superconducting walls have been found to have a Q factor as high as 109 at cryogenic temperatures [16]. In the present work, electromagnetic ¯eld analysis for the shielded dielectric resonators has been presented. Expressions for the ¯eld components, the characteristic equations and the quality factors have been derived. The resonant frequencies and the quality factors have been computed for the optical and microwave regions. 2. Theory The shielded homogeneous and isotropic spherical dielectric resonator is in principle equivalent to a spherical hole in a perfect conductor, ¯lled with the dielectric mate- rial. Waldron [3] has presented the analysis for a spherical cavity in a perfect con- ductor. However, the procedure followed by Waldron [3] seems to be rather clumsy. In the present work eigenmodes of a spherical homogeneous and isotropic dielectric resonator enclosed in a metallic spherical shell are determined using straightforward procedure. In a source-free homogeneous and dielectric medium the four Maxwell's equations are given by ~ D~ = 0; (1a) r ¢ ~ B~ = 0; (1b) r ¢ @B~ ~ E~ = ; (1c) r £ ¡ @t @D~ ~ H~ = : (1d) r £ @t Constitutive relations for B~ and D~ are given by B~ = ¹H~ = ¹0¹rH;~ (2a) D~ = "E~ = "r"0E~ ; (2b) where ¹0 and "0 are respectively the permeability and permittivity of the free space, ¹ and " are the corresponding quantities for the dielectric material and ¹r = ¹=¹0, "r = "="0. For a non-magnetic dielectric, ¹r = 1 and hence, B~ = ¹0H~ . 1256 Pramana { J. Phys., Vol. 62, No. 6, June 2004 Normal modes and quality factors of spherical dielectric resonators Assuming ej!t time dependence for E~ and H~ and using eqs (2a) and (2b), eqs (1a){(1d) reduce to ~ E~ = 0; (3a) r ¢ ~ H~ = 0; (3b) r ¢ ~ E~ = j!¹ H;~ (3c) r £ ¡ 0 ~ H~ = j!" " E:~ (3d) r £ r 0 Using expression for the curl in the spherical polar coordinates system (Appendix A, eq. (A1)) eqs (3c) and (3d) give 1 @(r sin θE ) @(rE ) j!¹ H = Á θ ; (4a) 0 r ¡r2 sin θ @θ ¡ @Á ½ ¾ 1 @(E ) @(r sin θE ) j!¹ H = r Á ; (4b) 0 θ ¡r sin θ @Á ¡ @r ½ ¾ 1 @(rE ) @(E ) j!¹ H = θ r ; (4c) 0 Á ¡ r @r ¡ @θ ½ ¾ 1 @(r sin θH ) @(rH ) j!" " E = Á θ ; (5a) r 0 r r2 sin θ @θ ¡ @Á ½ ¾ 1 @(H ) @(r sin θH ) j!" " E = r Á ; (5b) r 0 θ r sin θ @Á ¡ @r ½ ¾ 1 @(rH ) @(H ) j!" " E = θ r : (5c) r 0 Á r @r ¡ @θ ½ ¾ Equations (4a){(4c) can be used to ¯nd magnetic ¯eld components provided the ¯eld electric components are known. Similarly, eqs (5a){(5c) can be used to ¯nd ¯eld electric components provided the magnetic ¯eld components are known. Now taking the curl of eq. (3c) we get ~ (~ E~ ) = j!¹ (~ H~ ): (6) r £ r £ ¡ 0 r £ Using the vector identity (eq. (A2)), and using eqs (3a) and (3d) we get 2 + ¹ " " !2 E~ = 0: (7) r 0 0 r Similarly, taking¡ the curl of¢ eq. (3d) and using eqs (3b) and (3c) we get 2 + ¹ " " !2 H~ = 0: (8) r 0 0 r Equations (7)¡ and (8) represent¢ di®erential equations for the electric vector E~ and the magnetic vector H~ respectively. In the following we use the standard theory Pramana { J. Phys., Vol. 62, No. 6, June 2004 1257 R A Yadav and I D Singh [17{20] to ¯nd the electric and magnetic ¯elds for the TE and TM modes separately. For the TE mode the following condition is satis¯ed: ~r E~ = 0: (9) ¢ The electric ¯eld (E~ ) can be written in terms of the gradient of some scalar function à as E~ = ~r ~ Ã: (10) £ r Evidently, eq. (10) satis¯es condition (9) (one may verify this using eqs (9) and (10) and the vector identity (A3)). Here, à is any well-behaved scalar ¯eld that satis¯es the Helmholtz equation (Appendix B, eq. (B1)). Using à determined in the Appendix B (eq. (B11)) components of the electric ¯eld can be determined using eq. (10) and the expression of ~ à given in eq. (A4) as r Er = 0; (11a) mA E = J (kr)P m(cos θ) sin mÁ; (11b) θ pr sin θ n+(1=2) n A d E = J (kr) P m(cos θ) cos mÁ: (11c) Á pr n+(1=2) dθ f n g Alternatively, introducing the angular momentum operator L~ de¯ned as, L~ = (1=j)(~r ~ ), where j = p 1, and constructing L2 and its relationship with the Laplacian£ r operator ( 2), the¡ solution for TE mode can be constructed following Jackson [20]. Both ther methods yield equivalent results as can be veri¯ed from the ¯eld expressions (11a){(11c) and the ones given by Jackson [20]. Now substituting the values of Er; Eθ and EÁ from eqs (11a){(11c) into eqs (4a){ (4c), one obtains expressions for Hr; Hθ and HÁ. The RHS of eq. (4a) involves Eθ and EÁ and substituting the values of Eθ and EÁ from eqs (11b) and (11c) it yields AJn+(1=2)(kr) cos mÁ Hr = 3=2 ¡ j!¹0r d2 d sin2 θ P m(cos θ) 2 cos θ P m(cos θ) dθ2 f n g ¡ dθ f n g 2 : (12) £ 2 m m 3 2 Pn (cos θ) 6 ¡sin θ f g 7 4 5 m Using recurrence relations for Pn (cos θ) (Appendix C, eqs (C1) and (C2)) the term within the square bracket of eq. (12) is simpli¯ed to give n(n+1)P m(cos θ). ¡ n Therefore, the expression for Hr becomes n(n + 1)A m Hr = 3=2 Jn+(1=2)(kr)Pn (cos θ) cos mÁ: (12a) j!¹0r To get the expressions for Eθ and EÁ is straightforward, as RHSs of eqs (4b) and (4c) involve Er which vanishes for the TE mode leaving single term for these equations. The expressions for Eθ and EÁ are determined as 1258 Pramana { J. Phys., Vol. 62, No. 6, June 2004 Normal modes and quality factors of spherical dielectric resonators A d d m Hθ = prJn+(1=2)(kr) Pn (cos θ) cos mÁ; (12b) j!¹0r dr dθ f g © ª mA d m HÁ = prJn+(1=2)(kr) Pn (cos θ) sin mÁ: (12c) ¡j!¹0r sin θ dr © ª Similarly, for the TM modes the ¯eld components are given by n(n + 1)A m Er = 3=2 Jn+(1=2)(kr)Pn (cos θ) cos mÁ; (13a) j!"0"rr A d d m Eθ = prJn+(1=2)(kr) Pn (cos θ) cos mÁ; (13b) j!"0"rr dr dθ f g © ª mA d m EÁ = prJn+(1=2)(kr) Pn (cos θ) sin mÁ; (13c) ¡j!"0"rr sin θ dr © ª Hr = 0; (14a) mA H = J (kr)P m(cos θ) sin mÁ; (14b) θ ¡pr sin θ n+(1=2) n A d H = J (kr) P m(cos θ) cos mÁ: (14c) Á ¡pr n+(1=2) dθ f n g It must be mentioned here that the ¯eld expressions obtained in the present case di®er from those obtained by Waldron [3] by a factor of j!¹0 for the TE modes and by a factor of j!"0"r for the TM modes.
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