Orientation-dependent spontaneous emission rates of a two-level quantum emitter in any nanophotonic environment
aa r X i v : . [ phy s i c s . op ti c s ] D ec Orientation-dependent spontaneous emission rates of a two-level quantum emitter inany nanophotonic environment
Willem L. Vos , , ∗ A. Femius Koenderink , † and Ivan S. Nikolaev ‡ Center for Nanophotonics, FOM Institute for Atomic and Molecular Physics AMOLF, Amsterdam, The Netherlands and Complex Photonic Systems (COPS), MESA + Institute for Nanotechnology, University of Twente, The Netherlands (Dated: Published as Phys. Rev. A 80, 053802, 2009.)We study theoretically the spontaneous emission rate of a two-level quantum emitter in anynanophotonic system. We derive a general representation of the rate on the orientation of the tran-sition dipole by only invoking symmetry of the Green function. The rate depends quadratically onorientation and is determined by rates along three principal axes, which greatly simplifies visual-ization: Emission-rate surfaces provide insight on how preferred orientations for enhancement (orinhibition) depend on emission frequency and location, as shown for a mirror, a plasmonic sphere, ora photonic bandgap crystal. Moreover, insight is provided on novel means to ”switch” the emissionrates by actively controlling the orientation of the emitters’ transition dipole.
I. INTRODUCTION
It is well-known that the characteristics of sponta-neously emitted light depend strongly on the environ-ment of the light source [1, 2, 3, 4]. According to quan-tum electrodynamics, the emission rate of a two-levelquantum emitter, described by Fermi’s golden rule, isgenerally factorized into a part describing the sourcesintrinsic quantum properties and another part describ-ing the influence of the environment on the light field.Currently, there are many efforts to control the emissionrate of quantum emitters by optimizing the nanoscaleenvironment by, e.g. , reflecting interfaces [1, 5], micro-cavities [6, 7], photonic crystals [8, 9, 10, 11], or plas-monic nanoantennae [12, 13, 14]. Control of spontaneousemission is notably relevant to applications, includingsingle-photon sources for quantum information, minia-ture lasers and light-emitting diodes, and solar energyharvesting [15, 16, 17].The effect of the environment of a source on its emis-sion rate is described by the local density of optical states(LDOS) [4, 9, 11]. The LDOS counts the number of pho-ton modes available for emission, and it is interpretedas the density of vacuum fluctuations. In many exper-imentally relevant cases, it is theoretically known thatemission rates strongly differ for various orientations ofthe transition dipole moment see, e.g. , [4, 18]. Thus, thewidely pursued control of position and frequency leavesa large uncertainty in the emission rate [9]. To date, noclear picture has emerged of the general characteristicsof the orientation dependence. It is an open questionwhether the behavior mimics the local symmetry aroundthe emitter, see Fig. 1, or whether any generic depen-dence exists at all.Therefore, we present fundamental insights in the com-plex dependence of the emission rates of a quantum emit- ∗ † ‡ Now at ASML, Veldhoven, the Netherlands.
FIG. 1: (color online) Drawing of a two-level quantum emit-ter embedded in an arbitrary nanophotonic system, here de-picted as a cluster of 6 scatterers. If the emission rate wereto mimic the symmetry of the system, one would here expectan emission-rate surface with a 6-fold symmetry. Our analy-sis reveals, however, that these surfaces take on only specificshapes determined by the symmetry of the Green dyadic. Thesymmetry analysis allows one to conclude without any calcu-lation that the rate is identical for all dipole orientations inthe plane of the 6 scatterers. ter on the orientation of its dipole moment. Our general,yet simple theoretical analysis only invokes the symme-try of the Green function and provides a complete classi-fication of the orientation-dependences that the emissionrate can assume in any nanophotonic system. This classi-fication leads to an intuitive visualization that is based ononly a few clearly defined physical parameters, as shownby examples of an emitter near a mirror, a plasmonicsphere, or in a 3D photonic bandgap crystal. From ouranalysis, we conclude that control over the orientation ofthe transition dipole moment opens novel applications:If one can tune the orientation of an emitter, one can”switch” emission from inhibited to enhanced and viceversa . In the field of quantum information [19], atomicqubits that fly by nanophotonic systems could acquirecontrollable phase shifts by tuning their orientation rel-ative to the principal axes.
II. THEORYA. Derivation of emission-rate surface
The rate of spontaneous emission Γ of a two-level dipo-lar quantum emitter in the weak-coupling approximationis equal to [4, 9, 11]:Γ( ω, r , e d ) = πd ω ~ ǫ N ( r , ω, e d ) , (1)with ω the emission frequency, r the source’s position, e d the dipole orientation, d the modulus of the matrixelement of the transition dipole moment. N ( r , ω, e d ) isthe local density of optical states (LDOS) that equals: N ( ω, r , e d ) = 6 ωπc ( e Td · Im( G ( r , r , ω )) · e d ) , (2)with G ( r , r , ω ) the Green dyadic [4]. Eq. (1) reveals thewell-known fact that the emission rate depends on thefrequency and the position of the emitter. As is wellknown, Eq. (2) is also applicable to emission dynamicsinside dissipative optical media. In such media, the imag-inary part of the Green dyadic describes the total decayrate, i.e. , the sum of the radiative decay rate and the rateof quenching induced by the environment. Hence, the re-sults in this paper carry over straightaway to the decaydynamics of dipoles emitters in dissipative nanophotonicenvironments.A didactic example to illustrate the dependence ofemission rates on frequency, position and dipole orien-tation is that of a source near a perfect mirror, seeFig. 2(A), which can be understood from image dipoleanalysis [1, 3]. The rate depends strongly on the dipoleorientation e d : at small distances a dipole parallel tothe mirror has a vanishing emission rate, which canbe interpreted as due to destructive interference of thedipole with its oppositely oriented image. In contrast, adipole perpendicular to the mirror has twice the unper-turbed rate owing to constructive interference, as shownin Fig. 2(B). Clearly, the symmetry of this particular ge-ometry implies that the parallel and perpendicular dipoleorientations are ‘principal’ orientations along which themaximum and minimum rates are attained. At interme-diate orientations the rate is a weighted average of thetwo rates.The main result of our paper is that the rate always depends on orientation via a quadratic form with threeperpendicular principal axes, as will now be proven: Onaccount of reciprocity, the Green dyadic is equal to itstranspose upon exchanging the coordinates. HenceIm( G ( r , r ′ , ω )) T = Im( G ( r ′ , r , ω )) . (3)Furthermore, the imaginary part of the Green dyadic isreal. Therefore, the imaginary part of the Green dyadicin Eq. (2) is a real and symmetric 3 × ω and spatial position r , the imaginary part of the Green dyadic can always be diago-nalized, and has 3 eigenvalues ( g , g , g ) that correspondto three orthogonal eigenvectors. Since the eigenvaluescan be ordered by magnitude, we relabel the eigenvaluesand the concomitant main axes as { v min , v med , v max } .This basis corresponds to three perpendicular principaldipole orientations that vary with dipole location r andfrequency ω . In this orthonormal basis we express thedipole orientation unit vector e d as: e d = β v min + β v med + β v max , (4)where β i are coefficients that are constrained through β + β + β = 1 to lie on a unit sphere, since k e d k =1. Clearly, the coefficients β i are functions of the dipoleorientation: β i = β i ( e d ) = e Td · v i .Using Eqs. (1, 2), the emission rate Γ can be expressedin emission rate coefficients Γ i , which are the rates fordipole orientations parallel to the principal axes v i , lead-ing to:Γ( e d ) = β ( e d )Γ min + β ( e d )Γ med + β ( e d )Γ max . (5)Equation (5) describes the emission-rate surface as afunction of dipole orientation Γ( e d ), which is a centralresult of our work. The emission rate coefficients Γ i areequal to:Γ i = πd ω ~ ǫ ωπc ( v T i · Im( G ) · v i ) = πd ω ~ ǫ ωπc g i , (6)and are via G functions of the frequency and the dipoles’position: Γ i = Γ i ( ω, r ). Assuming known principal ratesΓ i , the emission-rate surface Γ( e d ) is always a quadraticform on the unit sphere. Moreover only quadratic formsof signature s = P (sign(Γ i )) = 3 can occur [20], sinceemission rates are physically constrained to be positivefor all orientations. Therefore, polar plots of the rateversus dipole orientation - henceforth called emission-rate surface - take on only specific shapes classified bythe ratios of Γ min , Γ med , Γ max , with three perpendicularsymmetry axes, regardless of the nanophotonic system.We remark that while Eq. (5) may appear as the definingequation of an ellipsoid, the emission-rate surface is not an ellipsoid since the problem is not about calculatinga level surface of Eq. (5), which would be equivalentto constraining β i to yield a fixed Γ in Eq. (5), ratherthan constraining β i to the unit sphere. Our result thatemission surfaces are always necessarily quadratic formsdefies the intuition (as sketched in Fig. 1) that emissionrates inherit the symmetry of the nanophotonic system.Regarding the assumptions we require to arrive at thequadratic form for the emission rate surfaces, we notethat we have assumed real dipole moment in Eq. (2) (fol-lowing Ref. [4]) and that we used reciprocity to ensurereal and symmetric Im( G ( r , r )). In case of reciprocalmedia it is easy to show that our results are also validfor complex transition dipole moments, and not just forreal dipole moments. Furthermore, if we assume a realdipole moment, it appears that our results are also valid FIG. 2: (color) (A)
Drawing of a two-level quantum emitter at distance h above a mirror. (B) Emission rate versus scaleddistance (wave vector times distance kh ) for a dipole perpendicular and parallel to a perfect mirror [3]. (C) Three-dimensionalsurfaces representing the orientation dependent spontaneous emission rate in real space. (i) One maximal emission and twoequal minimal rates give a peanut-shape (at kh = 0 . kh = 2 . kh = 3 . (D) Most general shape when allprincipal rates are different (Γ max > Γ med > Γ min ) and the principal axes are rotated from the ( x, y, z )-axes. Color scales arelinear from Γ min to Γ max (colorbar in (C)). for metamaterials that violate reciprocity, i.e., in caseIm( G ( r , r )) is not symmetric or even not diagonalizable.Since Im( G ( r , r )) is still real it will nonetheless give riseto a quadratic form that can be transformed to a prin-cipal axis system [20]. The physical requirement thatrates are positive for all dipole orientations furthermoreensures that the signature of the quadratic form remains3 even in the nonreciprocal case. B. Generic shapes of the emission-rate surface
Figure 2(C,D) categorizes all possible shapes of theemission rate polar plot. Fig. 2(C) is relevant for themirror, with principal axes parallel ( x, y , degenerate) and perpendicular ( z ) to the interface. Fig. 2(C(i)) showsthe emission-rate surface for the case where emissionis enhanced along a single dipole orientation Γ max ≫ Γ min = Γ med . This situation appears at a reduced dis-tance kh = 0 . min in the x, y -plane, and extendingto Γ max along the z -axis. Fig. 2(C(ii)) shows the ori-entation dependent emission rate for a single inhibitedaxis with Γ min ≤ Γ med = Γ max , at kh = 2 . kh = 3 . mostgeneral case when i) the rates along the main axes areall different (Γ min < Γ med < Γ max ), and ii) the princi-pal axes v min , med , max have an arbitrary orientation withrespect to the laboratory frame. Clearly, the emissionrate is not extremal for a dipole parallel to any of the( x, y, z )-axes. An important feature of the emission-ratesurfaces is that they allow for an easy inspection of boththe anisotropy of the emission rates, and of the favorabledipole orientations compared to the usual ( x, y, z )-axesin real space. III. EFFICIENT METHOD TO CALCULATEEMISSION-RATE SURFACES
In many cases of practical interest, neither the Green’sfunction G nor the principal axes { v i } are a-priori known. Often algorithms based on a summation over allphoton modes are used that only yield the rate Γ for tar-get orientations e d chosen as a priori input. Reconstruct-ing emission-rate surfaces as in Fig. 2 by a dense samplingof orientations is not viable with such algorithms, dueto prohibitive computation times. A poignant exampleis the calculation of emission rates in photonic crystalsthat requires a summation over up to 10 Bloch modes,the calculation of each of which requires diagonalizationof a 10 × matrix, even for a single dipole orienta-tion [11, 21]. A popular alternative method that can con-veniently yield the emission rate for a single orientationis the finite difference time domain (FDTD) simulationmethod [22]. However, it appears difficult to calculateoff-diagonal elements of the Green tensor. Since the var-ious field components are not calculated on identical gridpoints, FDTD does not truly yield a Green dyadic on awell-defined position r . Hence, even if an algorithm isknown to calculate rates at fixed orientations, it is un-clear how to find the principal axes and rates, since Im( G )is simply not available for diagonalization. In view of thecomputational cost of evaluating the radiative rate at asingle dipole orientation, the main problem is to find outfor how many and for which orientations the emissionrate must be calculated to completely and exactly char-acterize the emission-rate surfaces. Here we describe anefficient method to find principal emission rates and ori-entations by evaluating the LDOS at the least possiblenumber of input orientations.We use the well-known fact that any function on theunit sphere is conveniently expanded in spherical har-monics Y lm ( θ, φ ) = P lm (cos( θ )) e imφ . Since the emis-sion surface is a quadratic form, we can apply the well-known fact that all quadratic forms on the unit spherecan be represented exactly by an expansion containing only terms up to l = 2, so thatΓ( e d ) = X l =0 l X m = − l a lm P lm (cos( θ )) e imφ (7)An easy proof that no terms beyond l = 2 are needed isobtained by expressing the spherical harmonics in termsof cartesian coordinates, rather than polar coordinateson the unit sphere [20], or conversely by expressing thecoefficients β i in terms of polar coordinates relative to the { v i } axis system. This substition leads to a trigonomet-ric expansion for Γ( e d ) with terms that are quadratic incosines and sines of θ and of φ , see Appendix A, Eq. (A1).The expansion coefficients for the spherical harmonicexpansion are given by inner products a lm = h Γ( e d ) , Y lm i = Z π dφ Z π dθ Γ( e d ) Y lm ( θ, φ ) sin( θ ) , (8)similar to the coefficients appearing in discrete Fouriertransformations, but now for transformation on theunit sphere. Mohlenkamp has developed a fast Fouriertransform method to calculate the coefficients numeri-cally [23], which requires a sampling of rates Γ at a dis-crete set of orientations, similar to the numerical evalu-ation of discrete Fourier coefficients by the sampling ofa periodic function on a discrete set of points. In thisapproach, the integral expression (8) for the expansioncoefficients for expanding a function f is replaced by adiscrete weighted sum:ˆ a lm = X k w k f ( θ k , φ k ) Y lm ( θ k , φ k ) sin( θ k ) , (9)where k runs over the finite set of sampling points. Such adiscrete approximation to the expansion coefficients a lm is in fact exact for all functions f that are exactly equalto a finite series of spherical harmonics up to order l max if: i) the angles ( θ k , φ k ) are chosen as the roots of thebasis functions of order l = l max + 1, and ii) the w k are appropriate weights. In the present case l max = 2.Thus the special points are the 18 roots of the sphericalharmonics Y l,m ( θ, φ ) of order l = 3. Furthermore, onemay appreciate that the spherical harmonic transform isa simple Fourier transform over φ , and a Legendre trans-form over cos θ . The weights w k are hence the weightsappropriate for Gauss-Legendre quadratures of order 3.Explicitly, the 18 special points occur at azimuthal an-gles φ = mπ/ m = 0 , , . . .
5) and at polar angles θ = arccos( p / , π/ , arccos( − p / w only depend on θ , and are 5 / θ = arccos( ± p / / θ = π/
2. Since one half of the 18 points (seefigure 3) is antipodal to the other half, inversion symme-try of the emission rate means that the rate need onlybe evaluated for 9 dipole orientations in order to find thefull spherical harmonic expansion.
FIG. 3: (color online) Special orientations, i.e. , points on theunit sphere, for which the decay rate needs to be calculatedin order to fully reconstruct emission-rate surfaces. The blueand red points together are the roots of l = 3 spherical har-monics, corresponding to φ = mπ/ m = 0 , , , . . .
5) and θ = arccos( p / , π/ , − arccos( p / / / IV. RESULTS FOR SEVERALNANOPHOTONIC EXAMPLES
To illustrate our analysis, we discuss the emission dy-namics of a quantum emitter inside a photonic crystal,illustrated in Fig. 4(A,B). These complex systems haveextreme variations of the emission rate versus frequencyon account of a bandgap where emission is completelyinhibited [8]. To obtain the rate for an emitter of arbi-trary orientation in a Si inverse opal, we have calculatedthe LDOS for the 9 special orientations by summing overall Bloch eigenmodes [24]. The crystal has a first order‘pseudogap’ at reduced frequency 0.55, and a photonicbandgap from 0.852 to 0.891. Figure 4(B) shows theemission rate for a salient position in the unit cell ( cf.
Fig. 4(A)): the rate is anisotropic for frequencies nearthe pseudogap, since it differs for dipoles pointing in ei-ther x, y or the z direction, which are the cubic symmetryaxes of the crystal. One might be tempted to perceivethe behavior to be as simple as a mirror, since it is thesame for both x and y . However, a plot of the maxi-mum, medium, and minimum emission rates (Fig. 4(C))shows that this perception is completely wrong: Alreadyat low frequency up to the pseudogap, the emission rateis strongly anisotropic. While anisotropic behavior inthe long-wavelength limit may seem surprising, its originin electrostatic depolarization effects has been discussedbefore [26, 27]. The maximum rate occurs for dipole ori-entation e d = ( − , , / √
2, and is much larger than therate for any of the x, y, z orientations, whereas the mini- mum rate for e d = (1 , , / √ a/λ > .
6) up to the bandgap, the orienta-tion of maximum rate changes to e d = (0 , , max / Γ min ) isstrong with peaks up to 340. In this particular exam-ple, the high symmetry at this spatial position fixes allprincipal axes. To demonstrate the applicability of ourmethod to general, nonsymmetric, cases we have alsostudied low-symmetry positions at constant frequency,see Fig. 4(A). Again strong anisotropies occur, with themaximum-emission axis (or inhibition-axis) continuouslychanging direction as a function of source position. Weconclude that emission-rate surfaces provide a compactrepresentation of the rich behavior of the dependence ofthe emission rates on dipole orientation.We emphasize that our classification of emission dy-namics by means of emission-rate surfaces is by no meansrestricted to dielectric systems and can also be appliedto dissipative nanophotonics systems that are of moderninterest, such as plasmonic and metamaterial structures.Our analysis rests purely on the symmetry of the Greendyadic in Eq. (2), which in the presence of optical ab-sorption describes the total decay rate (radiative rateplus induced nonradiative rate) of a quantum emitter.As an example, we discuss the textbook case of an emit-ter near a plasmonic sphere [32, 33], using the knownGreen’s function [29] ( cf. Fig. 5(A)). Figure 5(C) showsthat the emission-rate surface for the total decay rate hasa donut-like shape (Γ min ≤ Γ med = Γ max ) with 16-foldenhanced rates for a dipole parallel to the surface, and5-fold enhanced for a perpendicular dipole. For a fixeddipole orientation [4, 18], the angular distribution of theradiated power reveals a well-known five-lobed structure(B). A comparison of (B) and (C) illustrates the maindifferences between radiation patterns and emission-ratesurfaces: radiation patterns are relevant to a single dipoleorientation and do not necessarily have any symmetry,or are free to follow any symmetry inherent in the envi-ronment. Emission-rate surfaces on the other hand arerelevant to all orientations and have a symmetry limitedby the quadratic form. FIG. 4: (color) Emission rate for a quantum emitter in a photonic bandgap crystal. (A)
Left: 1 / x = 0 . y = 0 .
3, variable z (red line in left panel)at reduced frequency a/λ = 0 .
94 ( a is lattice parameter). Surfaces are colored by relative rate (scalebar on right), and haveconstant size. (B) Emission rate for dipole centred in a window of a Si inverse opal [ r = 1 / , , e d = (1 , , , ,
0) (black curve), and (0 , ,
1) (blue dashed-dotted curve) versus a/λ . The rate is normalizedto the one in vacuum. The blue vertical bar indicates the photonic bandgap. Inset: cubic unit cell. (C)
Maximum, medium,and minimum emission rates Γ max (+), Γ med ( × ), Γ min ( | ) compared to rates for orientations e d = ( − , , / √ , ,
1) (dashed-dotted), (1 , , / √ (D) Emission-rate surfaces at select frequencies show strong changes inshape. The size of the surfaces is in proportion to the absolute emission rates, and colorscales range from Γ min to Γ max . Dashedcurve: Γ max .FIG. 5: (color) (A)
Drawing of a two-level quantum emitter at 20 nm distance from a plasmonic Ag sphere [28] with radius R = 80 nm. (B) Angular distribution of the radiated power versus solid angle Ω for a single dipole orientation parallel tothe surface; the pattern has a complex five-lobed structure. (C)
Emission-rate surface showing the emission rate versus dipoleorientation. The pattern has a donut-like shape.
V. DISCUSSION
Since the analysis in this paper is based on Im( G ( r , r ))it is strictly valid for the total decay rate modification in-duced by the nanophotonic environment. Explicitly, inthe case of losses our proof only holds for the sum of theradiative rate and the non-radiative rate (Γ rad +Γ nonrad ),and not for the radiative rate Γ rad separately. To analyzethe radiative emission-rate surfaces one would need to an-alyze the far-field integral of the radiated power (quantityin Fig. 5(B)) as a function of the source orientation. Apriori it is not at all clear that such radiative rate sur-faces need have a quadratic form. Indeed, we have notsucceeded in proving the quadratic form for the radiativerate in the lossless case by analysis of far-field integrals, i.e. , without identifying Γ rad = Γ tot and subsequentlyanalyzing Im( G ( r , r )). We have numerically calculatedradiative emission rate surfaces for many low-symmetrydissipative plasmon sphere clusters, and have not foundany example in which the radiative emission rate surfacewas not quadratic. Although a rigorous proof is beyondthe scope of this paper, we therefore anticipate that thequadratic form not only holds for total decay rates, butalso for radiative decay rates.A class of quantum emitters with a single transitiondipole moment are fluorescent molecules, such as laserdyes [4]. For such emitters, emission-rate surfaces can beobserved if their orientation is controlled, e.g. , by attach-ing them to liquid crystal molecules that are oriented inexternal fields [31]. If one can tune the orientation ofan emitter, this opens a novel opportunity to ”switch”spontaneous emission from inhibited to enhanced and vice versa . The emission-rate surfaces reveal that optimalswitching always requires a dipole rotation by 90 ◦ , sinceminimal and maximal emission rates always occur alongthe mutually perpendicular main axes. Alternatively, onecould tune semiconductor nanowires with oriented dipolemoments. For self-assembled and colloidal quantum dotswith dipoles in a x ′ , y ′ plane, we expect to probe the x ′ , y ′ cross-sectional average of the emission-rate surface of therelevant nanophotonic system.Since arbitrary orientations do not usually coincidewith principal dipole orientations, most prior work onspecific systems has been incomplete, since no principalrates has been reported. While such incompleteness doesnot affect the orientation averaged rate (see Appendix A),it does affect the understanding of dynamics of orienta-tional dipole ensembles [6, 10, 13]. Such a decay is a sumof single exponentials with a rate distribution given bythe emission-rate surface. Any observable derived fromtime-resolved decay beyond the orientation-averaged rate(Tr(Im( G ))) requires knowledge of the principal rates,which is thus relevant to many physical situations innanophotonics.In classical optics, the imaginary part of the Greendyadic is not only relevant for radiating dipoles. Indeed,the imaginary part of the Green dyadic has also been con-nected to the so-called coherency matrix (or the electric cross-spectral density tensor) [34] for black body radia-tion. In general, the 3 × levelsets (rather than polar plots) of an equation of the formin Eq. (5). It should be noted that the coherency matrixdepends on the incident source that generates the localelectric field. In the particular case that the field is dueto black body radiation the coherency matrix reduces tothe imaginary part of the Green dyadic Im( G ( r , r )), asderived by Set¨al¨a et al. [35]. However, it is importantto realize that for this identification of Im( G ( r , r )) withthe coherency matrix to hold, the source is required tobe a statistically homogeneous and isotropic distributionof radiating currents, and the medium is supposed to benon-dissipative [35]. This is diametrically opposite to theanalysis of spontaneous emission sources presented here,which concerns localized and oriented sources and is validwithout limitation on material dissipation. It is excitingthat our method to find principal rates and orientationscan be directly adapted to calculate the local polarizationproperties of black body radiation. VI. SUMMARY
We have theoretically studied the spontaneous emis-sion rate of a two-level quantum emitter in any nanopho-tonic system. We derive a general representation ofthe dependence of emission rates on the orientation ofthe transition dipole by only invoking symmetry of theGreen function. The rate depends quadratically on ori-entation and is determined by rates along three princi-pal axes. We show that these principal rates and axescan be easily calculated without evaluation of the fullGreen function. Furthermore we show that visualizationof emission-rate surfaces as determined from principalrates provides great insight on how preferred orientationsfor enhancement (or inhibition) depend on emission fre-quency and location, and on strategies to actively switchemission rates by the dipole orientation, as shown for amirror, a plasmonic sphere, or a photonic bandgap crys-tal.
VII. ACKNOWLEDGMENTS
We thank Allard Mosk, Ad Lagendijk, Peter Lodahlfor useful discussions. This work is part of the re-search program of the Stichting voor Fundamenteel On-derzoek der Materie (FOM) that is financially supportedby the Nederlandse Organisatie voor WetenschappelijkOnderzoek (NWO). WLV also thanks NWO-Vici andSTW/NanoNed.
APPENDIX A: DISCUSSION OF AVERAGEEMISSION RATE
A remarkable fact is that the orientation-average rate h Γ i can always be calculated from the LDOS at just threeperpendicular orientations, which need not coincide withthe principal axes { v min , v med , v max } . First, we calcu-late the orientation averaged rate by integration over thefull emission surface. Without loss of generality we align x, y, z with the principal axes, so that the orientation-dependent rate is:Γ( θ, φ ) = Γ min cos φ sin θ +Γ med sin φ sin θ +Γ max cos θ. (A1)By straightforward integration, the orientation-averagedrate h Γ i is h Γ i = 14 π Z π dφ Z π Γ( θ, φ ) sin θdθ = 13 (Γ min +Γ med +Γ max ) . (A2) Integration over the full emission surface clearly showsthat the orientation-averaged emission rate is equal tothe mean of the three principal rates, and hence h Γ i =( πd ω/ ~ ǫ )(2 ω/πc · Tr(Im( G ( r , r , ω ))). The invarianceof the trace of any matrix under arbitrary basis rotationimplies that the average rate in Eq. (A2) can be calcu-lated from the rates at any randomly chosen but mutuallyorthogonal directions x, y, z as h Γ i = 13 (Γ x + Γ y + Γ z ) . (A3) [1] K. H. Drexhage, J. Lumin. , 693 (1970).[2] D. Kleppner, Phys. Rev. Lett. , 233 (1981).[3] S. Haroche, in Fundamental systems in quantum optics ,Eds. J. Dalibard, J.M. Raimond, and J. Zinn-Justin(North Holland, Amsterdam, 1992), p. 767.[4] L. Novotny and B. Hecht,
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