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Forster resonance energy transfer rate in any dielectric nanophotonic medium withweak dispersion

Wubs, Martijn; Vos, Willem L.

Published in:New Journal of Physics

Link to article, DOI:10.1088/1367-2630/18/5/053037

Publication date:2016

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Wubs, M., & Vos, W. L. (2016). Forster resonance energy transfer rate in any dielectric nanophotonic mediumwith weak dispersion. New Journal of Physics, 18(5), [053037]. https://doi.org/10.1088/1367-2630/18/5/053037

arX

iv:1

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0621

2v1

[ph

ysic

s.op

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22

Jul 2

015

Forster resonance energy transfer rate and local density of optical states are

uncorrelated in any dielectric nanophotonic medium

Martijn Wubs∗

Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark andCenter for Nanostructured Graphene, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

Willem L. Vos†

Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology,University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

(Dated: July 22nd, 2015)

Motivated by the ongoing debate about nanophotonic control of Forster resonance energy transfer(FRET), notably by the local density of optical states (LDOS), we study an analytic model systemwherein a pair of ideal dipole emitters - donor and acceptor - exhibit energy transfer in the vicinity ofan ideal mirror. The FRET rate is controlled by the mirror up to distances comparable to the donor-acceptor distance, that is, the few-nanometer range. For vanishing distance, we find a completeinhibition or a four-fold enhancement, depending on dipole orientation. For mirror distances onthe wavelength scale, where the well-known ‘Drexhage’ modification of the spontaneous-emissionrate occurs, the FRET rate is constant. Hence there is no correlation between the Forster (ortotal) energy transfer rate and the LDOS. At any distance to the mirror, the total energy transferbetween a closely-spaced donor and acceptor is dominated by Forster transfer, i.e., by the staticdipole-dipole interaction that yields the characteristic inverse-sixth-power donor-acceptor distance inhomogeneous media. Generalizing to arbitrary inhomogeneous media with weak dispersion and weakabsorption in the frequency overlap range of donor and acceptor, we derive two main theoreticalresults. Firstly, the spatial dependence of the Forster energy transfer rate does not depend onfrequency, hence not on the LDOS. Secondly the FRET rate is expressed as a frequency integral of theimaginary part of the Green function. This leads to an approximate FRET rate in terms of the LDOSintegrated over a huge bandwidth from zero frequency to about 10× the donor emission frequency,corresponding to the vacuum-ultraviolet. Even then, the broadband LDOS hardly contributes to theenergy transfer rates. Using our analytical expressions, we plot transfer rates at an experimentallyrelevant emission wavelength λ = 628 nm that reveal nm-ranged distances, and discuss practicalconsequences including quantum information processing.

PACS numbers: 42.50.Ct, 42.50.Nn

I. INTRODUCTION

A well-known optical interaction between pairs ofquantum emitters - such as excited atoms, ions,molecules, or quantum dots - is Forster resonance en-ergy transfer (FRET). In this process, first identified ina seminal 1948 paper by Forster, one quantum of exci-tation energy is transferred from a first emitter, called adonor, to a second emitter that is referred to as an ac-ceptor [1]. FRET is the dominant energy transfer mecha-nism between emitters in nanometer proximity, since therate has a characteristic (rF /rda)

6 distance dependence(with rF the Forster radius and rda the distance betweendonor and acceptor). Other means to control a FRETsystem are traditionally the spectral properties of thecoupled emitters - the overlap between the donor’s emis-sion spectrum and the acceptor’s absorptions spectrum -or the relative orientations of the dipole moments [1, 2].FRET plays a central role in the photosynthetic appa-

∗Electronic address: mwubs@fotonik.dtu.dk†Electronic address: w.l.vos@utwente.nl, world wide web: www.photonicbandgaps.com

ratus of plants and bacteria [3, 4]. Many applicationsare based on FRET, ranging from photovoltaics [5, 6],lighting [7, 8], and magneto-optics [9], to sensing [10]where molecular distances [11, 12], and interactions areprobed [13, 14]. FRET is also relevant to the manipula-tion, storage, and transfer of quantum information [15–20].

Modern nanofabrication techniques have stimulatedthe relevant question whether Forster transfer can becontrolled purely by means of the nanophotonic environ-ment, while leaving the FRET pair geometrically andchemically unchanged. In many situations, the effect ofthe nanophotonic environment can be expressed in termsof the local density of optical states (LDOS) that countsthe number of photon modes available for emission, and isinterpreted as the density of vacuum fluctuations [21, 22].An assumption behind many recent FRET studies hastherefore been that if there is an effect of the nanopho-tonic environment on FRET rates, then it should be pos-sible to find a general law that describes the functionaldependence of FRET rates on the LDOS. While it is anassumption, it is a fruitful one as it allows for experi-mental verification. Curiously, different dependencies ofFRET rates on the LDOS were reported in a number

2

of experimental studies, leading to the debate how theForster energy transfer rate depends on the LDOS. Pio-neering work by Andrew and Barnes indicated that thetransfer rate depends linearly on the donor decay rateand thus the LDOS at the donor emission frequency [23],as was confirmed elsewhere [24]. The linear relation be-tween the two rates found in [23] was supported by thetheory of Ref. 25, but only fortuitously within a limitedparameter regime. Subsequent experiments suggested atransfer rate independent of the LDOS [27], a dependenceon the LDOS squared [26], or qualitative effects [28, 29].Possible reasons for the disparity in these observations in-clude insufficient control on the donor-acceptor distance,on incomplete pairing of every donor to only one accep-tor, or on cross-talk between neighboring donor-acceptorpairs.

Recently, the relation between Forster transfer and theLDOS was studied using precisely-defined, isolated, andefficient donor-acceptor pairs [30]. The distance betweendonor and acceptor molecules was fixed by covalentlybinding them to the opposite ends of a 15 basepair longdouble-stranded DNA. A precise control over the LDOSwas realized by positioning the donor-acceptor pairs atwell-defined distances to a metallic mirror [22, 31, 32].The outcome of this experimental study was that theForster transfer rate is independent on the optical LDOS,as was confirmed by theoretical considerations. Con-sequently, the Forster transfer efficiency is greatest fora vanishing LDOS, hence in a 3D photonic band gapcrystal [33]. Similar results were obtained with differentlight sources (rare-earth ions), and with different cavi-ties [34, 35]. On the other hand, a linear relation betweenLDOS and FRET rate was reported in experiments withdonors and acceptors at a few nanometers from metalsurfaces [36, 37], while Ref. [38] reported no general the-oretical relationship between LDOS and FRET rate neara metallic sphere. In Ref. 39 the measured dependenceof the FRET rate on the LDOS was reported to be weakfor single FRET pairs; an observed drop of the total en-ergy transfer rate of a donor close to a surface was mainlyattributed to the simple fact that fewer statistically dis-tributed acceptors are available close to the surface; re-cent theoretical work on collective energy transfer sup-ports these results in the dilute limit [40].

Most theoretical papers agree that both the energytransfer rate and the spontaneous-emission rate can beexpressed in terms of the Green function of the nanopho-tonic medium. One may argue that energy transfer andoptical LDOS are therefore related. But one may alsoargue to the contrary, since the energy transfer rate de-pends on the total Green function describing propaga-tion from donor to acceptor, whereas the spontaneous-emission rate depends on the imaginary part of the Greenfunction at the donor position only [25]. It is not clearwhether this situation entails correlations between thetwo quantities and if so, what is their functional rela-tionship. Therefore, and in view of the different resultsin the literature, we decided that a study of a simple

rda z

rda

parallel perpendicular

z

Figure 1: (Color online) We study pairs of donor and acceptordipoles that are separated by a distance rda, and located at adistance z from an ideal mirror. We focus on two configura-tions, both with dipoles oriented perpendicular to the positiondifference vector of donor and acceptor (µd, µa) ⊥ (rd − ra):(a) Both dipole moments of donor and acceptor are parallelto the mirror surface (‘parallel configuration’, ‖) and parallelto each other; (b) Both dipole moments of donor and accep-tor are perpendicular to the mirror surface (‘perpendicularconfiguration’, ⊥) and parallel to each other.

analytical model is timely.In this paper, we first study energy transfer in a pro-

totypical nanophotonic medium, namely the Drexhagegeometry [31], near an ideal mirror. This is one ofthe simplest inhomogeneous dielectric media for whichposition-dependent spontaneous-emission rates are ana-lytically known [32]. Here we show that the spatial de-pendence of both the total and the Forster resonant en-ergy transfer rates can be calculated analytically. Suchexact results have a value of their own, and allow for acritical and straightforward assessment of possible cor-relations between the FRET rate and the LDOS. Afterstudying the phenomenology near an ideal mirror, we de-rive general results for arbitrary inhomogeneous weaklydispersive media, and thereby find Forster transfer ratesthat generalize the well-known 1/(n4r6da) dependence ofthe homogeneous nondispersive medium with refractiveindex n. Specifically, using the energy-transfer theoryby Dung, Knoll, and Welsch [25] and the Green-functionproperties derived in Ref. [41] as starting points, we de-rive a simple and important new consequence: for thelarge class of photonic media that have little dispersionand absorption in the donor-acceptor frequency overlaprange, there is no correlation between the LDOS and theposition-dependent FRET rate. Nevertheless, the FRETrate is controlled by the distance to the mirror, but onlyat mirror distances comparable to or smaller than thedonor-acceptor separation.

II. PHYSICAL PROCESSES AND GEOMETRY

We study energy transfer from a single donor to a sin-gle acceptor separated by a distance rda = |ra − rd|. To

3

limit parameter space, we focus on situations in whichthe donor and the acceptor have the same distance z tothe mirror, and where the dipole moments of dipole andacceptor point in the same direction. In the parallel (‖)configuration shown in Figure 1(a), both dipole momentsare oriented parallel to the mirror, and the dipoles pointnormal to the mirror in the perpendicular (⊥) configura-tion of Figure 1(b).The total energy transfer rate γda between a donor and

an acceptor dipole in any nanophotonic environment isgiven by

γda =

∫ ∞

−∞

dω σa(ω)w(ra, rd, ω)σd(ω), (1)

where σd,a(ω) are the donor (single-photon) emissionand acceptor (single-photon) absorption spectra in freespace [25, 42]. All effects of the nanophotonic environ-ment are contained in the transfer amplitude squaredw(ra, rd, ω) that can be expressed in terms of the Greenfunction G(ra, rd, ω) of the medium, and the donor andacceptor dipole moments µd,µa respectively, as

w(ra, rd, ω) =2π

~2

(ω2

ε0c2

)2

|µ∗a · G(ra, rd, ω) · µd|

2. (2)

These expressions for the total energy transfer rate wereoriginally derived in an important paper by Dung, Knoll,and Welsch for a general class of nanophotonic mediathat may exhibit both frequency-dispersion and absorp-tion [25] [57]. The total energy transfer rate is the com-bined effect of both radiative and Forster energy transferprocesses. We will first study total energy transfer rates,and in Sec. IV we discuss what fraction of this energytransfer is Forster resonance energy transfer.For the energy transfer rate Eq. (1) we only need to

know the Green function in the frequency interval wherethe donor and acceptor spectra overlap appreciably. Incase of molecules that have among the broadest band-widths, this overlap has a typical relative bandwidthof only a few percent. Hence it is reasonable to ne-glect absorption and material dispersion in this narrowoverlap region. Thus ε(r, ω) can be approximated bya real-valued frequency-independent dielectric functionε(r). The corresponding Green function G(r, r′, ω) is thesolution of the usual wave equation for light

−∇×∇×G(r, r′, ω)+ ε(r)(ω

c

)2

G(r, r′, ω) = δ(r− r′)I,

(3)with a localized source on the right-hand side [58]. Un-like ε(r), the Green function G(r, r′, ω) is frequency-dependent and complex-valued.While the energy transfer rate in Eq. (1) evidently

depends on the donor and acceptor spectra σd(ω) andσa(ω), we are in this paper more interested in the de-pendence on the environment as given in Eq. (2). Wetherefore assume that the donor and acceptor overlap ina narrow-frequency region in which the transfer ampli-tude w(ra, rd, ω) varies negligibly with frequency, so we

Glossary of transfer and emission ratesγda total donor-acceptor energy transfer rate, Eq. (1)γda narrowband approximation of transfer rate, Eq. (4)γse spontaneous emission rate of the donor, Eq. (5)γF exact FRET rate from donor to acceptor, Eq. (21)

γ(L)F broadband LDOS approximated FRET rate, Eq. (27)

γ(HF)F high-frequency approximated FRET rate, Eq. (29)

Table I: Symbols for the various energy transfer and emissionrates used in this paper, with their defining equations.

can approximate the energy transfer rate by

γda = w(ra, rd, ωda)

∫ ∞

−∞

dω σa(ω)σd(ω), (4)

where ωda is the frequency where the integrand in theoverlap integral assumes its maximal value. The overlapintegral is the same for all nanophotonic environments,so that the ratio of energy transfer rates in two differentenvironments only depends on the ratio of w(ra, rd, ωda)in both environments.Spontaneous emission of the donor is a process that

competes with the energy transfer to the acceptor. Thedonor spontaneous-emission rate γse(r,Ω) at position r

with real-valued dipole moment µ = µµ and transitionfrequency ωd is expressed in terms of the imaginary partof the Green function of the medium as

γse(rd, ωd) = −

(2ω2

d

~ε0c2

)

µ · Im[G(rd, rd, ωd)] · µ (5)

or γ(rd, ωd,µ) = πµ2ωdρp(rd, ωd, µ)/(3~ε0) in termsof the partial LDOS ρp(rd, ωd, µ) = −(6ωd/πc

2)µ ·Im[G(rd, rd, ωd)] · µ, where µ is a dipole-orientation unitvector [21, 43]. The optical density of states (LDOS)is then defined as the dipole-orientation-averaged partialLDOS [43]. In general both the LDOS and the partialLDOS for any dipole orientation are fixed once the partialLDOS is known for nine independent dipole orientations,but for planar systems considered here, the two directions⊥ and ‖ suffice for a complete description [44] [59]. Wedo not average over dipole orientations, as we are inter-ested in possible correlations between energy transfer andspontaneous-emission rates for a fixed dipole orientation.

To proceed we need to compute Green functions. First,the Green tensor in a homogeneous medium with real-valued refractive index n is given by [46]

Gh(r1, r2, ω) = Gh(r, ω) =

−ew

4πr[P (w)I +Q(w)r ⊗ r] +

1

3(nω/c)2δ(r)I, (6)

where r = r1 − r2, the functions P,Q are defined asP (w) ≡ (1−w−1+w−2) and Q(w) ≡ (−1+3w−1−3w−2),and the argument equals w = (inωr/c). For n = 1,Gh equals the free-space Green function, denoted by G0.For distances much smaller than an optical wavelength

4

(r = |r| ≪ λ = 2πc/(nω)), the Green function scales asGh(r, ω) ∝ 1/(n2r3). From Eqs. (1) and (2) we then ob-tain the characteristic scaling of the Forster transfer rateas γda ∝ 1/(n4r6da): the Forster transfer rate stronglydecreases with increasing donor-acceptor distance andwith increasing refractive index. In contrast, it followsfrom Eq. (5) that the spontaneous-emission rate γse ina homogeneous medium is enhanced by a factor n com-pared to free space. More refined analyses that includelocal-field effects likewise predict a spontaneous-emissionenhancement [47]. These major differences between theenergy transfer rate and the emission rate in a homoge-neous medium already give an inkling on the behavior innanophotonics.

Next, we determine the Green function of an ideal flatmirror within an otherwise homogeneous medium withrefractive index n. While the function can be foundwith various methods [32, 50], we briefly show how itis obtained by generalizing the multiple-scattering for-malism of Ref. 48 for infinitely thin planes. In the usualmixed Fourier-real-space representation (k‖, z) relevantto planar systems with translational invariance in the(x, y)-directions, the homogeneous-medium Green func-tion Gh(k‖, z, z

′, ω) becomes

Gh =

1 0 00 k2z −k‖kzszz′

0 −k‖kzszz′ k2‖

c2

(nω)2gh+

δ(z − z′)

(nω/c)2zz,

(7)where the scalar Green function is given by gh =gh(k‖, z, z

′, ω) = exp(2ikz|z−z′|)/(2ikz), kz = (nω2/c2−

k2‖)1/2, szz′ = sign(z − z′) and the matrix is represented

in the basis (sk, pk, z), where k is the wave vector ofthe incoming light, z is the positive-z-direction, sk is thedirection of s-polarized light (out of the plane of inci-dence), and pk points perpendicular to z in the plane ofincidence. An infinitely thin plane at z = 0 that scatterslight can be described by a T-matrix T(k‖, ω), in termsof which the Green function becomes

G(z, z′) = Gh(z, z′) + Gh(z, 0)TGh(0, z

′), (8)

where the (k‖, ω) dependence was dropped. It was foundin Ref. 48 that for an infinitely thin plane that modelsa finite-thickness dielectric slab of dielectric constant ε,the T-matrix assumes a diagonal form in the same basisas Gh in Eq. (7), in particular T = diag(T ss, T pp, 0). Theinfinitely thin plane becomes a perfectly reflecting mir-ror if we choose for example a lossless Drude responsewith ε = 1−ω2

p/ω2, in the limit of an infinite plasma fre-

quency ωp → ∞. Hence the T-matrix for a perfect mirrorin a homogeneous dielectric has nonzero diagonal compo-nents T ss = −2ikz and T pp = −2i(nω/c)2/kz. The idealmirror divides space into two optically disconnected halfspaces, and below we only consider the half space z ≥ 0.It then follows that the Green function for the ideal mir-ror is written in terms of homogeneous-medium Green

functions as

G(z, z′) = Gh(z−z′)−Gh(z+z′)+2

(k‖c

nω

)2

gh(z+z′)zz,

(9)where the (k‖, ω)-dependence of the Green functions wasagain suppressed. To understand energy transfer ratesnear a mirror, we need to determine the Green functionin the real-space representation, which is related to theprevious equation by the inverse Fourier transform

G(r, r′, ω) =1

(2π)2

∫

d2k‖G(k‖, z, z′, ω)eik‖·(ρ−ρ

′),

(10)where ρ = (x,y) and ρ

′ = (x′,y′) so that r = (ρ, z).Knowing that this inverse Fourier transform when ap-plied to Gh(k‖, z − z′, ω) leads to the expression (6)also helps to evaluate the transform of Gh(k‖, z + z′, ω).Similarly, the third term on the right-hand side ofEq. (9) transform analogous to the zz-component of thehomogeneous-medium Green tensor. We thus find theGreen function for an ideal mirror within a homogeneousmedium as the sum of three terms:

G(r, r′, ω) = Gh(r, r′, ω)− Gh(ρ, z + z′,ρ′, 0, ω)

+2Gzz0 (ρ, z + z′,ρ′, 0, ω)zz. (11)

For the parallel configuration, we find

µ‖ ·G(ra, rd, ω) · µ

‖ = −µ2 einωrda/c

4πrdaP (inωrda/c)

+ µ2 einωu/c

4πuP (inωu/c), (12)

where rda is the donor-acceptor distance, z the distanceof both dipoles to the mirror, and u ≡ [r2da + (2z)2]1/2,

and µ‖ = µy as in Fig. 1. Inserting this result into

Eq. (2) immediately gives the squared transfer amplitudew(ra, rd, ω) of Eq. (2) for the parallel configuration.For the perpendicular configuration we find

µ⊥ · G(ra, rd, ω) · µ

⊥ = −µ2 einωrda/c

4πrdaP (inωrda/c)(13)

−µ2 einωu/c

4πu

[

P (inωu/c) + 4(z

u

)2

Q(inωu/c)

]

,

with µ⊥ = µz as in Fig. 1 and u as in Eq. (12), whereby

the squared transfer amplitude of Eq. (2) is also deter-mined for the perpendicular configuration.For completeness, we also give single-emitter

spontaneous-emission rates near the mirror (neglectinglocal-field effects [47] here and in the following). Fora dipole at a distance z and oriented parallel to themirror, we find from Eqs. (5) and (11)

γ‖se(z, ω) = γse,h(ω)

1−3

2

[sin(α)

α+

cos(α)

α2−

sin(α)

α3

]

,

(14a)

5

in terms of α = 2nωz/c and the homogeneous-mediumspontaneous-emission rate γse,h = µ2nω3/(3π~ε0c

3), i.e.,n times the spontaneous-emission rate in free space.For a dipole emitter oriented normal to the mirror, theposition-dependent spontaneous emission rate becomes

γ⊥se(z, ω) = γse,h(ω)

1− 3

[cosα

α2−

sinα

α3

]

. (14b)

In the limit z → 0, the rate γ‖se(z, ω) vanishes, while

γ⊥se(z, ω) tends to 2γse,h. In the limit z → ∞, both

γ‖se(z, ω) and γ⊥

se(z, ω) tend to the homogeneous-mediumrate γse,h(ω). These two expressions are well known [50].Here we see how these exact results also follow from ourmultiple-scattering approach; we will compare their spa-tial dependence with that of the analytically determinedenergy transfer rates obtained by the same approach.

III. ENERGY TRANSFER NEAR A MIRROR:PHENOMENOLOGY

Figure 2 shows the total energy transfer rate betweena donor and an acceptor as a function of distance z to themirror. In this figure and all others below we use indexn = 1. The panels show results for several donor-acceptorspacings rda = λ/100, λ/50, λ/25. In all cases, the to-tal energy transfer reveals a considerable dependence atshort range. In the limit of vanishing dipole-mirror dis-tance (z → 0), dipoles perpendicular to the mirror have afour-fold enhanced transfer rate compared to free space.The factor four can be understood from the well-knownmethod of image charges in electrodynamics: at a van-ishing distance, each image dipole enhance the field two-fold, and since energy transfer invokes two dipoles, thetotal result is a four-fold enhancement.With increasing dipole-mirror distance, the rate shows

a minimum at a characteristic distance that is remarkablyclose to the donor-acceptor spacing z ≃ rda. At largerdistances z > rda, the transfer rate converges to the ratein the homogeneous medium. In Appendix A it is shownthat this holds more generally: away from surfaces orother inhomogeneities, the FRET rate in an inhomoge-neous medium scales increasingly as ∝ 1/(n4r6da), with nthe refractive index surrounding the donor-acceptor pair.Figure 2 also shows that in the limit of vanishing dipole-mirror distance (z → 0), dipoles parallel to the mirrorhave an inhibited transfer rate. This result can also beunderstood from the method of image charges, since eachimage dipole provides complete destructive interferencein the limit of zero distance to the mirror.With increasing dipole-mirror distance, the rate in-

creases monotonously, and reaches half the free-spacerate at a characteristic distance that is also remark-ably close to the donor-acceptor spacing z ≃ rda. Atlarger distances z > rda, the transfer rate tends to thehomogeneous-medium rate. It is remarkable that even ina simple system studied here, a dramatic modification of

0 10 20

0

3rda= /100

Distance z (nm @ =628nm)

"150

327F

oers

ter_

mirr

or.O

PJ",

2015

-03-

27

0

3

_|_ mirror // mirror

rda= /50

Ener

gy tr

ansf

er ra

te (

da/

da,0)

0.0 0.1 0.20

3rda= /25

Distance to mirror ( z/c)

Figure 2: (Color online) Total energy transfer rate between adonor and an acceptor dipole, scaled to the free-space trans-fer rate, versus distance to the mirror, for the parallel andperpendicular configurations. The lower abscissa gives thedistance in scaled units, and the upper abscissa absolute dis-tances at a wavelength λ = (2π · 100)nm = 628nm. Fromtop to bottom the three panels correspond to donor-acceptorspacings rda = λ/100, λ/50, λ/25, where dipole-mirror dis-tances equal to rda are marked by vertical dotted lines (offscale in the lowest panel).

the energy transfer rate is feasible. In other words, wecan already conclude that the energy transfer rate be-tween donor and acceptor is controlled by the distanceto the mirror. The open question is whether this controlis mediated by the LDOS.Figure 3 shows typical distances that characterize the

distance dependent energy transfer rates in Figure 2 ver-sus donor-acceptor distance. For the perpendicular con-figuration we plot the distance where the transfer ratehas a minimum, and for the parallel case we plot the dis-tance where the transfer rate equals 1/2 of the free spacerate. Both characteristic distances increase linearly withthe donor-acceptor distance with near-unity slope. Thisbehavior confirms that the distance dependence of theenergy transfer rates in Figure 2 occurs on length scalescomparable to the donor-acceptor distance.

6

0.00 0.02 0.04

0.0 12.6 25.1

0.00

0.05

0.10

0.15

rda (nm @ =628nm)

"150

327F

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mirr

or.O

PJ",

2015

-03-

27

Cha

r. di

stan

ce (z

/)

Donor-acceptor distance (r da/ )

_|_ orient.: minimum // orient.: half height, linear

Figure 3: (Color online) Characteristic mirror separations z/λat which the total energy-transfer rate converges to the one infree-space. For dipoles perpendicular to the mirror, the char-acteristic distance is shown at which the total energy transferrate γDA/γDA,0 has a minimum (see Fig. 2). For dipoles par-allel to the mirror, the characteristic distance is shown whereγda/γda,0 equals 1/2 (see Fig. 2). The lines are linear fitsthrough the origin with slopes 3.75 and 3.45, respectively.

Figure 4 shows the distance-dependence of the energytransfer rate in comparison to the spontaneous-emissionrate. The latter varies with distance to the mirror onlength scales comparable to the wavelength of light, asfirst discovered by Drexhage [31]. In contrast, the en-ergy transfer rates vary on dramatically shorter lengthscales, about one-and-a-half (parallel configuration) totwo (perpendicular configuration) orders of magnitude

smaller than the wavelength scale.To graphically investigate a possible relation between

energy transfer rate and LDOS, Figure 5 shows a para-metric plot of the energy transfer rate as a function of(donor-only) spontaneous-emission rate, where each datapoint pertains to a certain distance z. The top abscissa isthe relative LDOS at the donor emission frequency thatequals the relative emission rate. The results at loweremission rate correspond mostly to the parallel dipoleconfigurations in Figs. 2 and 4, whereas the results athigher emission rate correspond to mostly to the perpen-dicular configurations in these figures. For three donor-acceptor distances (rda = λ/100, λ/50, λ/25) Figure 5shows that the energy transfer rate is independent of theemission rate and the LDOS over nearly the whole range,in agreement with conclusions of Refs. [27, 30, 34, 35].The energy transfer decreases fast near the low emissionrate edge and increases fast near the high emission rateedge, both of which correspond to distances very closeto the mirror (cf. Fig. 2). From Figure 5 it is readilyapparent that the energy transfer rate does not increaselinearly with the LDOS, leave alone quadratically, as pro-posed previously. The absence of a correlation betweenenergy transfer rate and LDOS that is phenomenologi-cally shown here is one of our main results, and will be

0.01 0.1 1 10

1 10 100 1000

0.0

0.5

1.0

1.5

2.0

2.5

0.0

0.5

1.0

1.5

2.0

2.5 Transf _|_ Transf //

rda= /100

Emission rate (

se /se,0 )

Distance z (nm @ =628nm)

"150

327F

oers

ter_

mirr

or.O

PJ",

2015

-03-

27

Ener

gy tr

ansf

er ra

te (

da/

da,0)

Distance to mirror ( z/c)

Emiss _|_ Emiss //

Figure 4: (Color online) Comparison of donor-acceptor en-ergy transfer rates γda and donor-only spontaneous emissionrates γse, as a function of the distance z to the mirror. Thelower abscissa is the scaled distance, the top abscissa is theabsolute distance for λ = 628 nm, both on a log scale. Theenergy transfer is scaled by the free-space energy transfer rateγda,0, the spontaneous emission by the free-space rate γse,0.Data are shown both for the parallel and for the perpendicularconfigurations. For vanishing distance, γda/γda,0 is inhibitedto 0 for the parallel and enhanced to 4 for the perpendicularconfiguration.

theoretically discussed in the remainder of this paper.

IV. ENERGY TRANSFER VERSUS LDOS

As is well-known, not all energy transfer is Forster en-ergy transfer. To find what fraction of the energy transfercorresponds to Forster energy transfer, we express theGreen function in terms of the complete set of opticaleigenmodes fλ that satisfy the wave equation

−∇×∇× fλ(r) + ε(r)(ωλ/c)2fλ(r) = 0, (15)

with positive eigenfrequencies ωλ > 0. The Green func-tion, being the solution of Eq. (3), can be expanded interms of these mode functions fλ. An important prop-erty of this expansion can now be obtained from Ref. 41(in particular by combining Eqs. (21) and (22) of [41]),namely that the Green function can be written as thefollowing sum of three terms:

G(r, r′, ω) = c2∑

λ

fλ(r)f∗λ(r

′)

(ω + iη)2 − ω2λ

︸ ︷︷ ︸

(16)

GR

−( c

ω

)2 ∑

λ

fλ(r)f∗λ(r

′)

︸ ︷︷ ︸

+(c/ω)2

ε(r)δ(r − r′)I.

GS

7

0 1 2

0

3rda= /100 _|_ mirror

// mirror

LDOS (Nrad/N0)

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27

0

3 LDOS Const.rda= /50

Ener

gy tr

ansf

er ra

te (

da/

da,0)

0 1 20

3rda= /25

Emission rate ( se/ se,0)

Figure 5: (Color online) Parametric plots of scaled energytransfer rate versus scaled spontaneous-emission rate, forthree donor-acceptor distances rda = λ/100, λ/50, λ/25 fromtop to bottom, respectively. Data are obtained from Fig. 4.Black horizontal lines are constant at γda/γda,0 = 1. Thegreen dashed lines are linear relations γda = γse.

Since the Green function controls the energy transfer rate(see Eq. (2)), it is relevant to discern energy transferprocesses corresponding to these terms. The first termin Eq. (16) denoted GR corresponds to resonant dipole-dipole interaction (RDDI), the radiative process by whichthe donor at position r emits a field that is then receivedby the acceptor at position r′. The name ‘resonant’ de-scribes that photon energies close to the donor and ac-ceptor resonance energy are the most probable energytransporters, in line with the denominator (ω+iη)2−ω2

λof this first term. The second term in (16) called GS cor-responds to the static dipole-dipole interaction (SDDI)that also causes energy transfer from donor to acceptor,yet by virtual intermediate processes (see also Sec. V.Bof Ref. 41). As explained below, it is this SDDI thatgives rise to the FRET rate that characteristically scalesas r−6

da in homogeneous media and dominates the total en-ergy transfer for strongly subwavelength donor-acceptorseparations. The third term in Eq. (2) is proportional to

the Dirac delta function δ(r− r′). Since r 6= r′ in case ofenergy transfer, this contribution vanishes.The fact that the Green function can be written as

the sum of three terms as in Eq. (16) is important, andimplies that for arbitrary environments the static part ofthe Green function can be obtained from the total Greenfunction by the following limiting procedure (for r 6= r′)

GS(r, r′, ω) =

1

ω2limω→0

ω2G(r, r′, ω), (17)

which provides a justification of our use of the term‘static’. As an important test, selecting in this waythe static part of the Green function of a homogeneousmedium (6) indeed gives that only

Gh,S(r1, r2, ω) =c20

4πn2ω2r3(I− 3rr) , (18)

with r = r1 − r2 contributes to Forster energy trans-fer, and not the terms of Gh that vary as 1/r and 1/r2.Incidentally and by contrast, for general inhomogeneousmedia the static Green function does not only depend onthe distance between donor and emitter, but rather onthe absolute positions of both donor and acceptor in themedium.Having thus defined Forster energy transfer as that

part of the total energy transfer that is mediated by thestatic dipole-dipole interaction, we can now also definethe square of the Forster transfer amplitude, in analogyto Eq. (2), by

wF(ra, rd, ω) =2π

~2

(ω2

ε0c2

)2

|µ∗a · GS(ra, rd, ω) · µd|

2.

(19)This appears similar to Eq. (2), yet with the total Greenfunction G replaced by its static part GS, as definedin Eq. (16) and computed in Eq. (17). The FRETrate γF is then obtained by substituting wF(ra, rd, ω) forw(ra, rd, ω) into Eq. (1), giving:

γF(ra, rd) =

∫ ∞

−∞

dω σa(ω)wF(ra, rd, ω)σd(ω). (20)

Here we arrive at an important simplification in the de-scription of Forster transfer in inhomogeneous media,by noting that from Eqs. (17) and (19), the quantitywF(ra, rd, ω) is independent of frequency ω. The FRETrate γF is then given by the simple relation

γF(ra, rd) = wF(ra, rd)

∫ ∞

−∞

dω σa(ω)σd(ω). (21)

While this expression looks similar to the approximateexpression for the total energy transfer rate (Eq. 4), weemphasize as a first point that Eq. (21) is an exact expres-sion for the FRET rate, even for broad donor and accep-tor spectra. A second crucial point is that the spectraloverlap integral in Eq. (21) is the same for any nanopho-tonic environment [60]. All effects of the nondispersive

8

inhomogeneous environment are therefore contained inthe frequency-independent prefactor wF(ra, rd). In otherwords, while there is an effect of the nanophotonic envi-ronment on the FRET rate (see Fig. 2), this effect de-pends only on the donor and acceptor positions but doesnot depend on the resonance frequencies of the donor andacceptor (for constant medium-independent overlap inte-gral in Eq. (21)). If the FRET rate does not depend onthe donor and acceptor frequencies, then the FRET ratecan not be a function of the LDOS at these particularfrequencies. A third crucial point is that this conclusionis valid for any photonic environment that is lossless andweakly dispersive in the frequency range where the donorand acceptor spectra overlap, hence this conclusion is notlimited to an ideal mirror.For homogeneous media it is well known that Forster

energy transfer dominates the total energy transfer atstrongly sub-wavelength distances, and we will now seethat this is also the case in inhomogeneous media, againtaking the ideal mirror as an example. The total energytransfer near an ideal mirror depends on the total Greenfunction as given in Eqs. (12) and (13) for the two dipoleconfigurations (cf. Fig. 1). For the donor and acceptornear the mirror in the parallel configuration, we use theprocedure of Eq. (17) and obtain for the static parts

µ‖·GS(ra, rd, ω)·µ

‖ =µ2c2

4πn2ω2

1

r3da−

1

(√

r2da + 4z2)3

,

(22)while for the perpendicular configuration we find

µ⊥ · GS(ra, rd, ω) · µ

⊥ =µ2c2

4πn2ω2

1

r3da+

1

(√

r2da + 4z2)3

(

1− 34z2

r2da + 4z2

)

. (23)

We note that in both cases the static interaction in a ho-mogeneous medium is recovered for FRET pairs at dis-tances to the mirror much larger than the donor-acceptordistance z ≫ rda. This agrees with the large-distancelimits shown in Figs. 2 and 4. The spatial dependenceof the Forster transfer amplitude of Eq. (19) and of theFRET rate in Eq. (21) is hereby determined for bothconfigurations.In Figure 6 we display the ratio of the FRET rate

and the total energy transfer rate as a function of donor-acceptor distance, for three distances z of the FRET pairto the mirror, and for both dipole configurations, takingn = 1. For the total rate we use the narrow bandwidthassumption of Eq. (4). Irrespective of the distance to themirror and of dipole configuration, the total energy trans-fer rate equals the FRET rate for all practical purposes,as long as (rdaω/c ≪ 1), in other words for strongly sub-wavelength donor-acceptor distances. Intriguingly, withincreasing donor-acceptor distance beyond typical FRETdistances, the ratio of the two rates exceeds unity. Sincethe total rate γda equals the absolute square of the sum of

10 100

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rda (nm @ =628nm)

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0.1 10

1dipoles _|_ mirror

Donor-acceptor distance ( rda/c)Fö

rste

r/ To

tal r

ate

(F/

DA)

Förster radiative

Figure 6: (Color online) Forster resonance energy transferrate scaled to the total energy transfer rate (γF/γda) versusdonor-acceptor distance rda for three distances z of donor andacceptor to the mirror. The upper panel is for dipoles par-allel to the mirror, the lower panel for dipoles perpendicularto the mirror. Note the logarithmic rda, with dimensionlessscaled values on the lower abscissa and absolute distance innanometers on the upper abscissa for λ = 628nm.

static and resonant transfer amplitudes, and the two am-plitudes interfere destructively in this intermediate dis-tance range, the FRET (static) rate can indeed dominatethe total energy transfer rate. At large donor-acceptordistances (rdaω/c ≫ 1), the FRET rate decreases muchfaster with distance than the total transfer rate, similaras in homogeneous media. In this large-distance range,the energy transfer is radiative: the donor emits a photonthat is absorbed by the acceptor.

Figure 7 is complementary to the previous one in thesense that here the FRET rate is plotted versus dis-tance to the mirror z for several donor-acceptor distancesrda, and for both dipole configurations. We again showthe ratio of the FRET rate and the total transfer rate(using Eq. (4) for γda). At donor-acceptor distancesrda = λ/100 and rda = λ/50, typical for experimentalsituations, we clearly see that FRET dominates the total

9

0 200 400

0.80.91.01.1

dipoles // mirror

Distance z (nm @ =628nm)

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090 2 4

1.0

1.1

rda= /100 rda= /50 rda= /20

dipoles _|_ mirror

Distance to mirror ( z/c)

FRET

rate

/ Tot

al ra

te (

F/da

)

Figure 7: (Color online) FRET rate γF divided by the totalenergy transfer rate γda, versus distance to the mirror, forthree values of the donor-acceptor distance rda. The lowerabscissa is the dimensionless reduced distance, the upper ab-scissa is the absolute distance in nanometer for λ = 628nm.Upper panel: parallel dipole configuration; lower panel: per-pendicular dipole configuration.

energy transfer rate, independent of the distance to themirror. At least 98% of the total energy transfer rate con-sists of the FRET rate. Even for a large donor-acceptordistance rda = λ/20 that is much larger than in mostexperimental cases (that is, rda = 31nm at λ = 628 nm),the FRET rate and the total rate differ by only someten percent. Thus, Figures 6 and 7 illustrate that inthe nanophotonic case near an ideal mirror, the FRETgreatly dominates the total energy transfer at stronglysub-wavelength donor-acceptor distances, similar as inthe well-known case of homogeneous media. Therefore,we conclude that not only is there no correlation betweenthe LDOS and the total transfer rate, there is also no cor-relation between the LDOS and the FRET rate either.

V. ENERGY TRANSFER IN TERMS OF AFREQUENCY-INTEGRATED LDOS

Our general derivation of the FRET rate in a weaklydispersive nanophotonic medium (Eq. 21) has convinc-ingly shown that the FRET rate has no dependence onthe local density of optical states evaluated at the donor’sresonance frequency. In this section we will insist on es-tablishing a link between the FRET rate and the LDOS,if only to counter the argument that we are from theoutset biased against such a relation. Interestingly, therelation that we derive will also shed a new light on effortsto control the FRET rate by LDOS engineering.We start with the mode expansion of the Green func-

tion in Eq. (16) to derive a useful new expression, relat-ing the Forster transfer rate to a frequency-integral overIm[G]. We use the fact that GS(r, r

′, ω) is real-valued,as is proven in Ref. [49]. Thus the imaginary part of theGreen function is equal to Im[GR] and the mode expan-sion of Im[G] becomes

Im[G(r, r′, ω)] = −πc2

2ω

∑

λ

fλ(r)f∗λ(r

′)δ(ω − ωλ), (24)

with ω > 0. We note that only modes with frequenciesωλ = ω show up in this mode expansion of Im[G]. Thiscan also be seen in another way: the defining equationfor the Green function Eq. (3) implies that the imaginary

part of the Green function satisfies the same source-freeequation (15) as the subset of modes fλ(r) for which theeigenfrequency ωλ equals ω. Therefore, Im[G(r, r′, ω)]can be completely expanded in terms of only those de-generate eigenmodes. The mode expansion (24) is indeeda solution of Eq. (15). If we multiply the right-hand sideof Eq. (24) with ω and then integrate over ω, we obtainas one of our major results an exact identity for the staticGreen function GS

GS(ra, rd, ω) =2

πω2

∫ ∞

0

dω1 ω1 Im[G(ra, rd, ω1)]. (25)

This identity is valid for a general nanophotonic mediumin which material dispersion can be neglected. Eq. (25)was derived using a complete set of modes, yet does notdepend on the specific set of modes used. When insertingthis identity into Eq. (19), we obtain the Forster transferrate wS(ω) and hence the transfer rate γF of Eq. (20)in terms of the imaginary part of the Green function.While this is somewhat analogous to the well-known ex-pression for the spontaneous-emission rate (Eq. 5), thereare two important differences: The first difference be-tween Eq. (25) for Forster energy transfer and Eq. (5)for spontaneous emission in terms of Im[G] is of coursethat Eq. (25) is an integral over all positive frequencies.The second main difference is that in Eq. (25) the Greenfunction Im[G(ra, rd, ω1)] appears with two position ar-guments - one for the donor and one for the acceptor -instead of only one position as in the spontaneous emis-sion rate. A major advantage of an expression in terms

10

of Im[G] is that Im[G] does not diverge for ra → rd, incontrast to Re[G].In Appendix B we verify and show explicitly that the

identity in Eq. (25) holds both in homogeneous mediaas well as for the nanophotonic case of arbitrary po-sitions near an ideal mirror. The identity in Eq. (25)is important since it is our goal in this section to ex-plore whether FRET rates are functionally related to theLDOS in any way. With Eq. (25), both quantities cannow be expressed in terms of the imaginary part of theGreen function, which brings us considerably closer tothe goal.We now use Eq. (25) to derive an approximate expres-

sion G(L)S for the static Green function GS that allows

us to relate the Forster transfer rate to the frequency-integrated LDOS. Our approximation is motivated bythe fact that Im[G(rd − ra, ω)] for homogeneous media(based on Eq. (6)) varies appreciably only for varia-tions in the donor-acceptor distance rda on the scale ofthe wavelength of light, typically rda ≃ λ0 = 500 nm(with λ0 = 2πc/ω0). From Eq. (11) it follows that thesame holds true for Im[G(rd, ra, ω)] for the ideal mir-ror. In contrast, Forster energy transfer occurs on alength scale of rda ≃ 5 nm, typically a hundred timessmaller. Motivated by these considerations, we approx-imate Im[G(ra, rd, ω1)] in the integrand of Eq. (25) bythe zeroth-order Taylor approximation Im[G(rd, rd, ω1)].The accuracy of this approximation depends on the opti-cal frequency ω. The approximation will not hold for allfrequencies that are integrated over, and becomes worsefor higher frequencies. But it appears that we can makean accurate approximation throughout a huge opticalbandwidth 0 ≤ ω1 ≤ Ω. If we choose Ω = 10ω0, i.e.,a frequency bandwidth all the way up to the vacuumultraviolet (VUV), then Im[G(ra, rd, ω1)] will only devi-ate appreciably from Im[G(rd, rd, ω1)] for donor-acceptordistances rda > λ0/10, which is in practice of the order of50 nm, much larger than typical donor-acceptor distancesin Forster transfer experiments. We obtain the expres-

sion for the approximate static Green function G(L)S as

G(L)S (ra, rd, ω) =

2

πω2

∫ Ω

0

dω1 ω1 Im[G(rd, rd, ω1)]

+2

πω2

∫ ∞

Ω

dω1 ω1 Im[G(ra, rd, ω1)].(26)

Here the first term is recognized to be an integral ofthe LDOS over a large frequency bandwidth, rangingfrom zero frequency (or ‘DC’) to high frequencies inthe VUV range. While the specific value of Ω doesnot matter much, it is important that Ω can be chosenmuch greater than optical frequencies, while the inequal-ity n(rd)Ωrda/c ≪ 1 still holds. Within this approxi-mation, the Forster transfer rate can be related to the

frequency-integrated LDOS: if we replace GS by G(L)S in

Eq. (19) for wF, thereby obtaining the LDOS approxi-

mation w(L)F (ra, rd, ω) = |µ∗

a · G(L)S (ra, rd, ω) · µd|

2, and

approximate wF in Eq. (20) by this w(L)F , we obtain the

0.9

1.0(a)

Min. wavelength min (nm)

dipoles // mirror

Appr

oxim

ate

FRET

/ För

ster

rate

(F(L

,HF)

/ F)

z= /100, rda= /100 z= /40, rda= /100 z= /2, rda= /100

628 62.8 31.4

0.9

1.0(b) dipoles _|_ mirror

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0 10 201.0

1.5 (c) z= /2, rda= /100

F(HF)

F(L)

LDOS bandwidth ( D)

Figure 8: (Color online) LDOS-approximated FRET rate γ(L)F

(Eq. (27)) normalized to the exact FRET rate γF (Eq. (20))versus the bandwidth Ω of the LDOS-frequency integral.Lower abscissa: Ω scaled by the donor frequency ωd = 2πc/λ.Upper abscissa: minimum wavelength λmin = 2πc/Ω forλ = 628nm. Black full curves are for dipole-to-mirror distancez = λ/100, red dashed curves for z = λ/40, and blue dashed-dotted curves for z = λ/2, all curves are for a donor-acceptordistance rda = λ/100. (a) Parallel dipole configuration; (b)perpendicular dipole configuration. (c) Comparison of theLDOS-approximation (Eq. (27)) and the high-frequency ap-proximation (Eq. (28)) of the FRET rate as a function ofLDOS bandwidth Ω. Rates are scaled to the exact FRETrate, and the distance to the mirror and the donor-acceptordistance are fixed.

relation between the approximate FRET rate γ(L)F and

the LDOS to be [61]

γ(L)F =

∫ ∞

−∞

dω σa(ω)w(L)F (ω)σd(ω). (27)

In Figure 8 we verify the accuracy of the LDOS-

approximated FRET rate γ(L)F near the ideal mirror, by

varying the frequency bandwidth Ω over which we makethe approximation. The required frequency integrals ofEq. (26) are calculated analytically in Appendix C 2. InFig. 8 we see that for both dipole configurations, the ap-

11

0 2 4

0 200 400

0.996

0.998

1.000

Distance z (nm @ =628nm)

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Rel

ativ

e ra

te (

F(L) /

F)

Distance to mirror ( z/c)

rda= /100 =10 D

= 2 D

Figure 9: (Color online) LDOS-approximated FRET rate γ(L)F

(Eq. (27)) normalized to the exact FRET rate γF (Eq. (20))versus (scaled) distance to the mirror, for an LDOS frequencybandwidth up to Ω = 10ω0 (red dashed curve), and up toΩ = 2ω0 (blue dashed-dotted curve). The donor-acceptordistance is rda = λ/100.

proximate FRET rate indeed tends to the exact rate forvanishing Ω. For Ω up to 10ωd, the approximate rate isvery close to the exact one, to within 5%. Even at higherfrequencies, up to Ω = 20ωd, the approximate FRETrate is within 10% of the exact rate, as anticipated onthe basis of our general considerations above.

The validity of the approximate FRET rate γ(L)F im-

proves when the donor-acceptor distance rda is reduced,since the spatial zero-order Taylor expansion of Im[G] isthen a better approximation. We can also improve theapproximation by reducing the frequency bandwidth Ω inwhich we make the Taylor approximation. Both trends

are indeed found in Appendix C 1 where γ(L)F is calculated

for the homogeneous medium. In the limit of a vanishingfrequency bandwidth (Ω → 0), the approximate Forster

transfer rate γ(L)F reduces to the exact Forster transfer

rate γF of Eq. (20).To verify that the approximate FRET rates shown in

Fig. 8 were not ‘lucky shots’ for the chosen fixed distancesto the mirror, we study in the complementary Figure 9(a)

the accuracy of γ(L)F as a function of distance to the mir-

ror z, for a constant LDOS bandwidth Ω = 10ωd. Thefigure clearly shows the great accuracy of the LDOS ap-proximation, irrespective of the distance z of the FRETpair to the ideal mirror. For a narrower bandwidth ofΩ = 2ωd, the accuracy is even better, as expected.At this point, one may naively conclude from Figures 8

and 9 that the FRET rate is intimately related to an in-tegral over the LDOS. This is too rash, however, becausethe corresponding approximate relation Eq. (27) consistsof two integrals, where only one of them is an integralover the LDOS, while the other is a high-frequency in-tegral of the complex part of the Green function. Thusthe relevant question becomes: what happens if we make

a cruder approximation to the FRET rate by simply re-moving the LDOS integral? Instead of Eq. (26) we thenuse the high-frequency approximation (HF) to the static

Green function G(HF)S :

G(HF)S (ra, rd, ω) =

2

πω2

∫ ∞

Ω

dω1 ω1 Im[G(ra, rd, ω1)].

(28)This leads to a high-frequency approximation

for the squared Forster amplitude w(HF)F (ω) =

(2π/~2)(ω/(ε0c2))2|µ∗

a · G(HF)S (ra, rd, ω) · µd|

2, and

a high-frequency approximation the FRET rate γ(HF)F :

γ(HF)F =

∫ ∞

−∞

dω σa(ω)w(HF)F (ra, rd, ω)σd(ω). (29)

In Figure 8(c) the two approximated FRET rates γ(L)F

and γ(HF)F are compared for the ideal mirror, both scaled

by the exact FRET rate γF, as a function of the band-width Ω. The donor-acceptor distance and the distance

to the mirror are fixed. Indeed γ(L)F is the more accurate

approximation of the two, yet γ(HF)F is not a bad ap-

proximation at all: by only integrating in Eq. (28) over

high frequencies ω1 ≥ Ω = 10ωd, γ(HF)F is accurate to

within about 7%. If we take a narrower – yet still broad– frequency bandwidth, for example up to Ω = 2ωd (inthe UV), we still neglect the LDOS in the whole visiblerange. Nevertheless Figure 8(c) shows that for Ω = 2ωd

the two approximations γ(L)F and γ

(HF)F agree to a high ac-

curacy with the exact rate γF. Therefore, Figures 8 and9 show that for the ideal mirror there is essentially no de-pendence of the FRET rate on the frequency-integratedLDOS at visible frequencies, and only a weak dependenceon the frequency-integrated LDOS at UV frequencies andbeyond. We note that this conclusion is different fromand complementary to the one in Sec. IV, where theFRET rate was found to not depend on the LDOS atone frequency, namely at the transferred energy ~ωd.

VI. DISCUSSION

In this section, we discuss consequences of our theoreti-cal results to experiments, first regarding relevant lengthscales. We have performed analytical calculations andplotted rates versus reduced lengths, namely the reduceddistance to the mirror zω/c = 2πz/λ, and the reduceddonor-acceptor distance rda/λ. To increase the relevanceof our results to experiments and applications, we haveplotted in several figures additional abscissae for abso-lute length scales that pertain to a particular choice ofthe donor emission wavelength λd. Here we have chosenλd = 2π/ωd = (2π · 100)nm ≃ 628 nm, a figure that werefer to as a ”Mermin-wavelength” [53], as it simplifiesthe conversion between reduced units and real units to amere multiplication by 100×.

12

Figures 2 and 3 characterize the distance dependenceto the mirror. The range where both the total and theFRET rate are controlled by the distance to the mirror isin the range z < 20 nm. This range is set by the donor-acceptor distance that is for most typical FRET pairs inthe order of rda = 10nm, in view of typical Forster dis-tances of the same magnitude [2]. Interestingly, while theenergy transfer is in this range (z < rda) not controlledby the LDOS, the transfer rate itself is nevertheless con-trolled by precise positioning near a mirror. An exampleof a method that could be used to achieve such controlat optical wavelengths is by attaching emitters - such asmolecules or quantum dots - to the ends of brush poly-mers with sub-10 nm lengths [54]. With Rydberg atoms,it appears to be feasible to realize the situation z < rda,albeit in the GHz frequency range [55].

Figure 6 characterizes the donor-acceptor distance de-pendence of the transfer rate. It is apparent that Forstertransfer dominates in the range rda < 20nm, a lengthscale much smaller than the wavelength of light. In therange rda > 100 nm, energy transfer is dominated byradiative transfer, which is reasonable as this distancerange becomes of the order of the wavelength.

Let us now briefly discuss broadband LDOS control. Ifone insists on invoking the LDOS to control the FRETrate, Figure 8 shows that one must control the LDOSover a huge bandwidth that ranges all the way from zerofrequency (’DC’) to a frequency Ω that is on the order of10 times the donor emission frequency ωd. This agreeswith the qualitative statements in Ref. [27]. If we con-sider the Mermin-wavelength 628 nm, the upper boundon the LDOS bandwidth corresponds to a wavelengthof 62.8nm nm, which is deep in the vacuum ultraviolet(VUV) range. At these very short wavelengths, all mate-rials that are commonly used in nanophotonic control - beit dielectrics such as silica, semiconductors such as silica,or metals such as silver or gold - are strongly absorbing.In practice, the optical properties of common nanopho-tonic materials deviate from their commonly used prop-erties at wavelengths below 200 to 250 nm, which cor-responds to Ω < 3ωd. Yet, even if one were able tocontrol the LDOS over a phenomenally broad bandwidth0 < Ω < 3ωd, Figure 9 shows that the broadband LDOS-integral contributes negligibly - much less than 10−3 - tothe Forster transfer rate. Thus, with the current state ofthe art in nanofabrication, true LDOS-control of Forsterenergy transfer seems to be extremely challenging.

The importance of distinguishing FRET from otherenergy-transfer mechanisms has also been emphasizedby Govorov and co-workers [51], who studied plasmon-enhanced Forster transfer near conducting surfaces.They predict that near metal surfaces the FRET ratecan be enhanced by much more than the factor 4 reportedhere for an ideal mirror, but that a strong enhancementoccurs only near the plasmon peak, i.e., in a highly dis-persive region, while on the other hand not too close tothe resonance since otherwise loss makes Forster trans-fer invisible. Analogous FRET enhancements only in the

highly dispersive region near a resonance were found inRef. [25] for single-resonance Drude-Lorentz type media.In Ref. [52], FRET near graphene is also clearly distin-guished from long-range plasmon-assisted energy trans-fer. As a future extension of our work, it will be inter-esting to study plasmon-enhanced FRET and long-rangeenergy transfer rates by considering a mirror with a res-onance.

How do our theoretical results compare to experi-ments? Our theoretical findings support the FRET-rateand spontaneous-emission rate measurements by Blum etal. [30], where it was found that Forster transfer rates areunaffected by the LDOS. Our findings also agree with theresults of Refs. [27, 34, 35, 39]. What about other exper-imental studies that do report a relation between FRETrate and LDOS? Assuming our theory to be correct, thisdiscrepancy can mean three things. First, it could bethat in those experiments the energy transfer betweendonor and acceptor separated by a few nanometers wasnot dominated by Forster transfer. This might occurfor energy transfer within a high-Q cavity, but otherwisedoes not seem to be a probable explanation. Second, ourtheory leaves open the possibility that there is a corre-lation between FRET rates and LDOS in case of strongdispersion and/or loss in the frequency overlap range ofdonor and acceptor spectra, in case of plasmon-mediatedFRET for example, because in that case our theory doesnot apply. However, if the correlation between LDOSand FRET rates indeed relies on strong dispersion, thenthis correlation would be a particular relation rather thanthe sought general relation. Moreover, Ref. 38 theoreti-cally studies spontaneous-emission and FRET rates nearmetal surfaces but does not report a general linear rela-tion between them. Technically, in case of non-negligibleabsorption and concomitant complex dielectric function,the concept of the LDOS breaks down, but FRET ratescan still be compared to the imaginary part of the Greenfunction. As a third possible reason why our theory doesnot predict a relation between FRET rates and LDOSwhile some experiments do, we should mention that ourtheory does not include typical aspects of experiments,such as incompletely paired donors, cross-talk betweendense donor-acceptor pairs, inhomogeneously distributeddonor-acceptor distances, or transfer rates influenced bystrongly inhomogeneous field distributions that may oc-cur near nanoparticles or slits and indentations. Whiledescribing these effects adds substantial complexity toour simple model, they may be taken into account byjudicious choices of simple scatterers such as spheres ordipoles. Nevertheless, it does not seem likely that suchparticular additional effects will induce a general depen-dence of the FRET rate on the LDOS.

Regarding the subject of quantum information process-ing, FRET is a mechanism by which nearby (< 10 nm)qubits may interact [15–20], intended or not. Lovett et

al. [17] considered the implications of Forster resonanceenergy transfer between two quantum dots. In one im-plementation, it was found that it is desirable to sup-

13

press the Forster interaction in order to create entangle-ment using biexcitons. In another implementation, it wasfound that Forster resonance energy transfer should notbe suppressed, but rather switched. There is a growinginterest in manipulating the LDOS, either suppressing itby means of a complete 3D photonic band gap [33], orby ultrafast switching in the time-domain [56]. It fol-lows from our present results that these tools cannot beused to also switch or suppress Forster resonance energytransfer between quantum bits. On the other hand, ourresults do indicate that FRET-related quantum informa-tion processing may be controlled by carefully positioningthe interacting quantum systems (i.e., the quantum dots)in engineered inhomogeneous dielectric environments.

VII. CONCLUSIONS

Using an exactly solvable analytical model, we haveseen that Forster resonance energy transfer rate from adonor to an acceptor differs from the one in a homoge-neous medium in close vicinity of an ideal mirror. Fortwo particular dipole configurations, we found that theFRET transfer rate is inhibited (to 0) or markedly en-hanced (4×). Thus, even this simple model system of-fers the opportunity to control energy transfer rates. Itturns out that differences in FRET rates as comparedto those in a homogeneous medium are only noticeableat distances to the mirror on the order of the donor-acceptor distance rda or smaller. This distance togetherwith the wavelength are the only natural length scalesin this simple problem. Since rda is typically less than10 nm, that is, orders of magnitude smaller than an op-tical wavelength, any substantial variations in the FRETrates due to the nanophotonic environment occur on adistance scale on which the LDOS does not vary appre-ciably in the dielectric medium near the ideal mirrror.On the larger distance scale of an optical wavelength

away from the mirror, there are well-known LDOS vari-ations. At these larger distances, the FRET rate is con-stant and the same as in a homogeneous medium. So theone quantity varies appreciably when the other does not,and vice versa. Therefore, we conclude that the FRETrate does not correlate with the partial or total LDOS.This particular example already suffices to conclude thatin general, the FRET rate does not correlate - neither lin-early, nor quadratically, or otherwise - with the LDOS.How large is the class of environments for which FRET

rate and LDOS are uncorrelated? We have derived asone of our main results the simple and exact expres-sion (Eq. 21) for the FRET rate in an inhomogeneousmedium. As a consequence, it follows that Forster en-ergy transfer rates are independent of the LDOS at thetransferred photon frequency in all nanophotonic mediawhere material dispersion and loss can be neglected in thedonor-acceptor frequency overlap interval. Other mainresults are the exact expression (Eq. 25) of the staticGreen function in terms of a frequency integral over the

imaginary part of the total Green function, and the corre-sponding approximate relation (Eq. 27) between FRETrate and the frequency-integrated LDOS. We used thelatter relation to show that FRET rates near an idealmirror are numerically independent of the LDOS, evenwhen integrating the LDOS over all visible frequencies.We have also argued why the same will be true for othermedia with weak material dispersion as well.We have emphasized that not all energy transfer is

Forster energy transfer, and that for typical Forsterdonor-acceptor distances below 10 nm, the energy trans-fer is dominated by Forster transfer, as the example ofthe ideal mirror has also shown. For arbitrary photonicenvironments we defined Forster transfer as being medi-ated by virtual photon exchange, the strength of whichis determined by the static Green tensor, which in homo-geneous media gives rise to the characteristic 1/(n4r6da)-dependence of the Forster resonance energy transfer rate.

Acknowledgments

It is a pleasure to thank Bill Barnes, Christian Blum,Ad Lagendijk, Asger Mortensen, and Allard Mosk forstimulating discussions, and Bill Barnes for pointing outRef. [53]. MW gratefully acknowledges support from theVillum Foundation via the VKR Centre of ExcellenceNATEC-II and from the Danish Council for IndependentResearch (FNU 1323-00087). The Center for Nanostruc-tured Graphene is sponsored by the Danish National Re-search Foundation, Project DNRF58. WLV gratefullyacknowledges support from FOM, NWO, STW, and theApplied Nanophotonics (ANP) section of the MESA+Institute.

Appendix A: Scaling with donor-acceptor distanceof Forster transfer rate

Here we show that the homogeneous-medium Forstertransfer rate, scaling as ∝ 1/(n4

hr6da), is an important

limiting case also for inhomogeneous media. Let us as-sume that the donor and acceptor are separated by afew nanometers, experiencing the same dielectric mate-rial with a dielectric function εh, within an inhomoge-neous nanophotonic environment. In all of space, we de-fine the optical potential V(r, ω) = −[ε(r) − εh](ω/c)

2I,

so that the optical potential vanishes in the vicinity ofthe donor-acceptor pair. Then the Green function of themedium can be expressed in terms of the homogeneous-medium Green function and the optical potential as

G(ra, rd, ω) = Gh(ra − rd, ω) (A1)

+

∫

dr1 Gh(ra − r1, ω) ·V(r1, ω) ·G(r1, rd, ω),

which is the Dyson-Schwinger equation for the Greenfunction that controls the energy transfer. The equa-tion can be formally solved in terms of the T-matrix of

14

the medium as

G(ra, rd) = Gh(ra − rd) (A2)

+

∫

dr1dr2 Gh(ra − r1) ·T(r1, r2) ·Gh(r2 − rD),

where the frequency dependence was dropped for read-ability. The important property of the T-matrixT(r1, r2, ω) is now that it is only non-vanishing whereboth V(r1) and V(r2) are nonzero, so that it vanishes inthe vicinity of the donor-acceptor pair. Thus the Greenfunction that controls the energy transfer is given by thesum of a homogeneous-medium Green function and ascattering term. The former is a function of the distancebetween donor and acceptor, whereas the latter does notdepend on the D-A distance, but rather on the distanceof donor and acceptor to points in space where the opticalpotential is non-vanishing.As the donor-acceptor distance rda is decreased, the

homogeneous-medium contribution in Eq. (A2) growsrapidly, essentially becoming equal to Gh,S(ra − rd, ω)of Eq. (18), whereas the contribution of the scatteringterm does not change much. So in the limit of very smallrda, or when making the distance to interfaces larger, thehomogeneous-medium term always wins, and one wouldfind the well-known Forster transfer rate of the infinitehomogeneous medium ∝ 1/(n4

hr6da).

Appendix B: Tests of identity (25)

1. Test for a homogeneous medium

The Green function Gh(r, ω) for homogeneous mediais given in Eq. (6), and its static part by Eq. (18). Theidentity (25) that relates them can be shown to hold asa tensorial identity; here we derive the identity for itsprojection µ ·Gh(r, ω) ·µ, where we assume µ to be per-pendicular to r. (Physically, this corresponds to energytransfer between donor and acceptor with equal dipolesboth pointing perpendicular to their position differencevector.) The projection of the identity (25) that we areto derive has the form

µ2c2

4πn2ω2

1

r3da= (B1)

µ2

2π2n2ω2

1

rda

∫ ∞

0

dω1ω1Im[

einωrda/cP (inωrda/c)]

.

Now by integration variable transformation the right-hand side of this equation can be worked out to give

µ2c2

2π2n2ω2r3da

∫ ∞

0

dx

[

cos(kx) + x sin(kx)−sin(x)

x

]

,

(B2)with dummy variable k equal to unity. Now thefirst two terms within the square brackets do not con-tribute to the integral since

∫∞

0dx cos(kx) = πδ(k) and

∫∞

0 dxx sin(kx) = −π ddkδ(k), while the third term in

the square brackets of Eq. (B2) does contribute since∫∞

0 dx sin(x)/x = π/2. Thus the projection of the iden-tity (25) indeed holds for spatially homogeneous media.

2. Test for an ideal mirror

For the ideal mirror we again only consider a projec-tion of the identity (25), first projecting onto dipolescorresponding to the parallel configuration of Fig. 1.The Green function for the ideal mirror is given inEq. (11), and its static part for the parallel configurationby Eq. (22). Now for this parallel configuration, the pro-jected Green tensor consists of a homogeneous-mediumand a reflected part, and so does the projected staticGreen function. In Sec. B 1 above we already showed thatthe sought identity indeed holds for homogeneous media.So the remaining task is to show that the identity (25)holds separately for the reflected parts of the projectedGreen functions. This is not difficult since mathemati-cally the frequency integral that is to be performed is thesame as for the homogeneous medium; only the distanceparameter rda is to be replaced by

√

r2da + 4z2. Thus theprojection of the identity (25) onto the parallel dipoledirections indeed holds. The qualitative novelty as com-pared to the homogeneous-medium case is that we thusshow that the identity holds irrespective of the distancez of the FRET pair to the mirror. We also checked (notshown) that the identity (25) holds for the projectiononto perpendicular dipoles, i.e. as in the perpendicularconfiguration of Fig. 1.

Appendix C: Accuracy of the approximateexpressions (26) and (29)

To test the accuracy of the LDOS approximation

G(L)S (ra, rd, ω) of the static Green function, it is conve-

nient to use Eq. (25) to rewrite Eq. (26) as

G(L)S (ra, rd, ω) = GS(ra, rd, ω) (C1)

+2

πω2

∫ Ω

0

dω1 ω1 Im[G(rd, rd, ω1)− G(ra, rd, ω1)],

In this form, the approximate static Green function isequal to the exact expression plus an integral over a fi-nite interval of a well-behaved integrand. Likewise, totest the accuracy of the high-frequency approximation

G(HF)S (ra, rd, ω) defined in Eq. (29) of the static Green

function, it is useful to rewrite it as

G(HF)S (ra, rd, ω) = GS(ra, rd, ω) (C2)

−2

πω2

∫ Ω

0

dω1 ω1 Im[G(ra, rd, ω1)],

Again the integrand is well-defined, i.e. non-diverging,over the entire finite integration interval.

15

1. Accuracy of LDOS approximation forhomogeneous media

We estimate the accuracy of Eq. (C1) for the Greenfunction (6) of a homogeneous medium. By taking theprojection onto dipole vectors both on the left and right,we find

Im

∫ Ω

0

dω1 ω1 µ · [Gh(rd, rd, ω1)− Gh(ra, rd, ω1)] · µ =

−µ2n

4πc

∫ Ω

0

dω1ω21

[2

3− h(D)

]

, (C3)

where D = nωd/c and for convenience we defined thefunction h(x) ≡ cos(x)/x2 + sin(x)(1− 1/x2)/x. So here(and also for the mirror below) we must determine inte-grals of the type

H(Ω, a) =

∫ Ω

0

dω1ω21h(ω1a) (C4)

= (Ω/A)3 [sin(A)− A cos(A)− Si(A)] .

where A = Ωa and Si[x] =∫ x

0dt sin(t)/t is the sine inte-

gral. For Ωa ≪ 1 we find the approximation

H(Ω, a) =2

9Ω3 −

2

75

(Ωa)5

a3. (C5)

With this result, we find that the relative error of making

the LDOS approximation G(L)h,S(ra, rd, ω) of Eq. (C1) for

the Green function Gh,S(ra, rd, ω) is

µ ·(

G(L)h,S − Gh,S

)

· µ

µ ·Gh,S · µ= −

4

75π(Ωrdan/c)

5. (C6)

This fifth-power dependence shows that for homogeneousmedia the LDOS approximation is excellent as long asΩrdan/c ≪ 1, which for typical Forster distances of afew nanometers corresponds to a frequency bandwidth Ωof order 10ωd in which the LDOS approximation can bemade, where ωd is a typical optical frequency (e.g., thedonor emission frequency).

2. Accuracy of LDOS approximation for the idealmirror

For the parallel configuration near the ideal mirror, wefind Eq. (C3), but with the integrand on the right-handside replaced by

−µ2n

4πcω21

[2

3− h(D1)

]

− [h(2Z1)− h(U1)]

, (C7)

where D1 = ω1rdan/c, Z1 = ω1zn/c, and U1 =√

D21 + 4Z2

1 . So we can identify both a homogeneous-medium and a scattering contribution between the curlybrackets. By threefold use of the identity (C4) it then

follows that µ · (G(L)S − GS) · µ equals

−µ2n

2π2ω2c

[2

9Ω3 −H(Ω,

nrdac

)

]

−

[

H(Ω,2nz

c)−H(Ω,

nu

c)

]

,

(C8)

where u =√

r2da + 4z2.

For the perpendicular configuration near the ideal mir-ror, it can be found that the integrand of Eq. (C3) isinstead replaced by the slightly longer expression

−µ2n

4πcω21

(2

3− 2h(2Z1) + 2

sin(2Z1)

2Z1

)

− h(D1)

−h(U1)−4z2

√

r2da + 4z2

[

2sin(U1)

U1− 3h(U1)

]

.(C9)

The frequency integral can again be performed usingthe identity (C4) and a standard integral of the type∫dxx sin(x). The resulting expression for µ · (G

(L)S −

GS) ·µ, and the corresponding result (C8) for the paral-lel configuration are both used in Figs. 8 and 9.

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[57] Our function w is the same as w in Ref. [25], Eq. (44).[58] Our definition of the Green function agrees with Refs. [41,

46] and differs by a minus sign from Refs. [25, 43].[59] What is for convenience called LDOS should in the fol-

lowing be understood to be the partial LDOS.[60] Let us recall here that σa(ω) and σd(ω) are the donor’s

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[61] In the symbol γ(L)F the superscript L is meant to indicate

a FRET rate in terms of the LDOS.