Local-field effects on spontaneous emission of impurity ... - COPS

10 January 2000Ž .Physics Letters A 264 2000 472–477www.elsevier.nlrlocaterphysleta<strong>Local</strong>-<strong>field</strong> <strong>effects</strong> on spontaneous emission of impurity atoms inhomogeneous dielectricsFrank J.P. Schuurmans ) , Pedro de Vries 1 , Ad LagendijkVan der Waals-Zeeman Instituut, UniÕersiteit Õan Amsterdam, Valckenierstraat 65, NL-1018 XE Amsterdam, the NetherlandsReceived 8 November 1999; accepted 29 November 1999Communicated by L.J. ShamAbstractThe local-<strong>field</strong> corrections to the spontaneous emission rate of an impurity atom in an otherwise homogeneous dielectricare calculated by making use of a macroscopic approach. An extension of a method introduced by Onsager and Bottcher ¨ ispresented. By distinguishing between the substitutional and interstitial character of impurities, the well-known empty-cavityand Lorentz local-<strong>field</strong> factors are obtained, respectively. For fluids, the substitutional case is predicted to occur prevalently.q 2000 Elsevier Science B.V. All rights reserved.PACS: 32.80.-t; 41.20.-q; 42.25.Bs; 78.55.-mKeywords: Modification of spontaneous emission; <strong>Local</strong>-<strong>field</strong> corrections; DielectricsIn 1917, Einstein demonstrated that spontaneousemission must occur if matter and radiation are toachieve thermal equilibrium wx 1 . This picture is sofundamental, that spontaneous emission rates areoften believed to be an inherent property of the atom.However, the spontaneous emission rate can be modifiedby changing the environment, as was noticed byPurcell wx 2 . In the early 1970s, Drexhage carried outpioneering experiments on the modification of theluminescence decay rate of Europium complexes infront of a metallic mirror w3,4 x. Later, many theoretiw5,6xand experimental w7–11xinvestigations cal have) Corresponding author. Tel.: q31-20-5255663, fax: q31-20-5255788, e-mail: schuurma@phys.uva.nl1 Present address: KNMI, Royal Netherlands MeteorologicalInstitute, P.O. Box 201, NL-3730 AE De Bilt, the Netherlandsbeen devoted to the enhancement and inhibition ofspontaneous emission of atoms in resonant cavities,in which case larger modifications are possible.Modifications of the spontaneous emission ratecan also be induced by placing the radiator inside aspatially inhomogeneous dielectric. For example, thecase of emission near a dielectric interface has beenstudied both experimentally w3,4,12–14xand theoretiw15,16x. More recently, spontaneous emissioncallyrates were predicted to alter in photonic crystalsw17–19x Žsee also for overviews Ref. w20 x. , materialsin which the dielectric constant is periodically varyingon length scales of the wavelength of light. As aresult, light in a certain frequency range cannotpropagate in the crystal because of multiple Braggreflections. At this so-called full photonic band gap,the radiative density of states Ž RDOS.is zero andfluorescence of excited atoms embedded in these0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00855-5

( )F.J.P. Schuurmans et al.rPhysics Letters A 264 2000 472–477 473materials is expected to be inhibited at those frequencies,since then coupling to propagating modesis absent. At other frequencies, it has been shownthat spontaneous emission can be either enhanced orreduced w17–19x Žsee also for overviews Ref. w20 x. .Note that the atoms in question are different from theones constituting the dielectric; such atoms will bereferred to as impurity atoms or, equivalently, impuritydipoles.The spontaneous emission rate G Ž ´ . of an impuritydipole inside a homogeneous medium with dielectricconstant ´ Žat the frequency of the impurityradiation.was predicted by Nienhuis and Alkemadew21xto follow the ' ´ dependence of the RDOS, andis thus given byŽ . 'Ž .G ´ s ´G , 10where G0is radiative decay rate of the dipole invacuum. This result was obtained by quantizing themacroscopic Maxwell’s equations, while in principalthe dipole couples to the microscopic vacuum fluctuations.The microscopic, or local, electric <strong>field</strong> feltby a dipole can be very different from the macroscopic<strong>field</strong> Žsee for example Ref. w22 x. , and hencelarge deviations from Eq. Ž. 1 may be expected.Determining the local <strong>field</strong> is an old problem inphysics, for which Lorentz has given a mean-<strong>field</strong>likeargument in terms of a ‘virtual’ cavity surroundw22,23x. For randoming a dipole of the dielectricfluids and cubic lattices, the resulting ratio of thelocal to the macroscopic <strong>field</strong> is given by the Lorentzlocal-<strong>field</strong> factor´q2LLors , Ž 2.3which is the same for all dipoles that constitute thedielectric. Indeed, in experiments w24xinvolving onlypure systems, i.e. when only one kind of atom ormolecule is present, the observed local-<strong>field</strong> <strong>effects</strong>agree very well with this Lorentz description. However,in the case of an impurity dipole, it is not at allclear whether Eq. Ž. 2 should apply.An alternative local-<strong>field</strong> correction that has beensuggested w25,26xis the so-called empty-cavity factorw22x3´Lemps , Ž 3.2´q1since the impurity would represent an excluded volumefor the dipoles constituting the dielectric. In adescription of G Ž ´ . of impurity dipoles inside dielectrics,one of the factors, LLoror L emp, shouldŽ . ' 2be included, giving G ´ s ´ L G w26–33xLor 0 orŽ . ' 2G ´ s ´ L G w25,26 x. Since G Ž ´ .emp 0can be ex-pressed in terms of an expectation value of theproduct of two electric <strong>field</strong> operators w26 x, thelocal-<strong>field</strong> factors in G Ž ´ . appear squared. The maw27–33xfavor thejority of the researchers in the <strong>field</strong>Lorentz factor. On the other hand, experimental rew34–36xindicate that the empty-cavity factorsultsw25xshould be employed.Recently, we have shown w37xthat for substitutionaland interstitial impurities, the empty-cavityand Lorentz-local <strong>field</strong> factor apply, respectively.This crucial distinction between substitutional andinterstitial character of the impurity was not made sofar. Our results were obtained from a full microscopicscattering calculation using a Green’s funcw38x.The aim of this Letter is to rederive the results oftion formalismRef. w37xin a much simpler way. First, the local-<strong>field</strong>factors for both types of impurities are calculatedemploying a quasi-static macroscopic approach. Thiscalculation is in the spirit of the method used byOnsager and Bottcher ¨ to derive the local-<strong>field</strong> forpure systems w23,39 x. Their method is extended hereto incorporate both substitutional and interstitial impurities.Second, the polarizability is expressed interms of a dynamic complex quantity of which theparameters are linked to one another by invoking theoptical theorem, valid for elastic light scattering. Inthis fashion, we obtain the appropriate local-<strong>field</strong>corrections for the spontaneous emission rate of theimpurities.We first investigate the effective static polarizabilityof the substitutional impurity. An impurity dipolewith polarizability a I is placed at the center of aspherical cavity with volume V in an otherwisehomogeneous medium with dielectric constant ´.The local-<strong>field</strong> correction pertaining to the <strong>field</strong> inthe cavity is then given by w23x3´LOB Ž a IrV . s . Ž 42a.I2´q1y Ž ´y1.3V

( )F.J.P. Schuurmans et al.rPhysics Letters A 264 2000 472–477 475arguments relating absorption, stimulated and spontaneousemission. The corresponding dynamic polarw38xizability of the impurity in Õacuum is v 02I I 2 2 3 2v 0yv yi G0v rv 0a Ž v. sa Ž 0 . , Ž 10.Ž .Here, a Ž.I 0 is the static polarizability, v 0 the resonancefrequency, and G0the resonance width, whichis identical to the spontaneous emission rate. The seta Ž. 0,v ,G 4I 0 0 has only two independent parameters,as the optical theorem relates them via w38xG0sa IŽ 0. v 0 4 r6p c03 , Ž 11.where c0is the speed of light in vacuum. Thisrelation is consistent with Fermi’s Golden Rule resultfor G by taking the static polarizability a Ž.0 I 0s2< D< 2 r´ 0"v0 in terms of the dipole matrix elementD of two atomic levels. These relations holdfor a dipole in vacuum. When considering a dipole ina background medium with dielectric constant ´ andneglecting local-<strong>field</strong> or cavity <strong>effects</strong>, Eq. Ž 11.ismodified to G Ž ´ . s ' ´G, 0 with G0determined bythe polarizability and the resonance frequency as inEq. Ž 11 .. This is in agreement with the Nienhuis andAlkemade result, see Eq. Ž. 1 .Implementing the dynamic polarizability a Ž v.Ifor the static one a in Eq. Ž.I 6 readily gives theeffective dynamic polarizability a Ž v.sub,I of a substi-tutional impurityasub,IŽ v.y12aIŽ v .Ž ´y1.L2empsaIŽ v.Lemp1y .9´VŽ 12.The polarizability of this ‘dressed’ impurity nearresonance has a Lorentzian structure analogous toEq. Ž 10 .:asub,I Ž v. sasub ,I Ž 0.v 02 Ž ´ .=,2 2 3 2v 0Ž ´ . yv yiŽ G Ž ´ . v rv 0Ž ´ ..Ž 13.with parameters given by2v 0Ž ´ . 2a IŽ 0 .Ž ´y1.Lemps 1y2v 0 9´V, Ž 14.v 02 2asub,I Ž 0. sLemp a Ž 0 ., 2 Iv Ž ´ .Ž 15.2G ´ v 0 ´' 2emp 2G0 v 00Ž . Ž .s ´ L . Ž 16.Here, the spontaneous emission rate G Ž ´ . of thesubstitutional impurity follows from the polarizabilitya Ž. 0 and the resonance frequency v Ž ´ .sub,I 0 byuse of the optical theorem, see Eq. Ž 11 .. It should benoted that a direct calculation of G Ž ´ . from Eq. Ž 13.is not feasible, as the macroscopic derivation presentedhere is in essence quasi-static, and hence onlyaccounts for a real polarizability. This drawback isnot encountered in the full, but more complicatedcalculation of Ref. w37 x. In vacuum, ´s1, the originalresults are regained, that is v Ž. 1 sv , a Ž.0 0 sub,I 0sa Ž. 0 , and G Ž.I 1 sG 0. The resonance frequency isred-shifted and the line width behaves in accordancewith the empty-cavity description, identical to ourearlier results w37 x.The dynamical properties of an interstitial impurityare obtained in a similar fashion as for thesubstitutional one. As a result, the polarizability ofthe ‘dressed’ interstitial impurity also has aLorentzian line shape given by Eq. Ž 13 .. The parametersfor the interstitial case are2v 0Ž ´ . 2a IŽ 0 .Ž ´y1.LLors 1y2v 0 9´V, Ž 17.v 02 2aint ,I Ž 0. sLLor a Ž 0 ., 2 Iv Ž ´ .Ž 18.2G ´ v 0 ´' 2Lor 2G0 v 00Ž . Ž .s ´ L . Ž 19.Again the resonance frequency is red-shifted, althoughwith a different functional dependence on ´.The Lorentz local-<strong>field</strong> factor is found to determinethe modification of the spontaneous emission rate, inagreement with Ref. w37 x.

476( )F.J.P. Schuurmans et al.rPhysics Letters A 264 2000 472–477The obtained results for both the substitutionaland interstitial impurity still depend on the volumeof the cavity V. For the case of the interstitials they1volume is fixed: Vsr Ž Onsager’s assumption ..For substitutionals, V is generally taken to be thereal volume of the impurity atom or molecule. Forsmall impurity volumes, V-r y1 , only one dipoleof the dielectric is expelled from the cavity. If theimpurity is large, V4r y1 , many dipoles of thedielectric are removed, and the impurity is a truemacroscopic cavity.The resonance frequency of the impurity slightlyshifts to the red, see Fig. 1. This shift is a directconsequence of the reaction <strong>field</strong> produced by thepolarization in the medium that is induced by theimpurity dipole. For small impurity polarization densitiesŽa Ž 0. rV