Collective F\"orster energy transfer modified by the planar metallic mirror

1.2 MB·English

by  Alexander N. Poddubny

Collective Fo¨rster energy transfer modified by the planar metallic mirror
Alexander N. Poddubny
Ioffe Institute, St Petersburg 194021, Russia and
ITMO University, St Petersburg 199034, Russia∗
(Dated: January 28, 2015)
We present a theory of the Fo¨rster energy transfer between the arrays of donor and acceptor
moleculeslyingontheplanarmetallicmirror. Werevealstrongmodificationoftheeffectivetransfer
ratebythemirrorintheincoherentpumpingregime. Theratecanbeeithersuppressedorenhanced
dependingontherelativepositionsbetweenacceptoranddonorarrays. Thestrongmodificationof
the transfer rate is a collective effect, mediated by the light-induced coupling between the donors;
5
it is absent in the single donor model.
1
0
PACSnumbers: 78.67.Pt,78.67.-n,33.50.-j,42.50.Nn
2

n
a I. INTRODUCTION
J

7
F¨orster energy transfer processes are now actively
2
studied in various fields that bridge physics, biology and

] medicine. The energy is transferred from the excited
l (donor) system to the acceptor one via the electromag-
l
a netic interaction. The donor and acceptor systems may
h
be realized as quantum dots and quantum wells,1–3 bi-
-
s ological molecules,4,5 defects in semiconductor.6,7 Typi-
e
cally, the range of the F¨orster interaction is on the order
m
of several nm.8
.
t One can try to control the efficiency of the trans-
a
fer by embedding the donors and acceptors into the
m
structured electromagnetic environment. Contradicting
- experimental reports of the enhanced9, modulated10,
d
n independent11,12 and suppressed13 transfer are available FIG. 1: Schematic illustration of the studied system. Arrays
o inliterature. IntheparticularcaseofRef.13thetransfer ofdonor(D)andacceptor(A)moleculesabovethesubstrate
c has been studied for the dye molecules embedded in the are indicated.
[ thinpolymerfilm. Thespeedupofthedonoremissionde-

1 cay in the presence of the acceptors has been attributed
v to the energy transfer rate. The transfer rate has been
In the rest of the paper we present the model (Sec. II)
7 suppressed when the film was put on top of the metallic
forthecollectivetransferandshow,thatthetransferrate
2 mirror. As has been pointed out in Ref. 12, for the pla-
8 can be strongly modified by the mirror (Sec. III).
narmirrorthiseffectcannotbeexplainedbytheexisting
6
well-developed theory of energy transfer between indi-
0
vidualindependentpairsofdonorsandacceptors.14–17 It
.
1 has been suggested in Ref. 13, that the observed trans-
0
fer suppression can be due to the collective effects, since
5
the concentration of the donor dye molecules was quite II. MODEL
1
: large and the intermolecular distance was on the order
v
of several nm. However, any further theoretical expla-
i The system under consideration is schematically de-
X nation was missing in Ref. 13. Developing a theory of
picted in Fig. 1. It consists of the metallic substrate
collectiveenergytransferfromdonorstoacceptorsinthe
r with the permittivity ε bounded from top with the
a presenceofaplanarmirroristhemaingoalofthecurrent subs
semi-infinite dielectric matrix with the permittivity ε .
study. Recently, similar theory has been put forward by b
Arrays of donor and acceptor molecules form square lat-
Pustovit,UrbasandShahbazyaninRef.18formolecules
ticeswiththeperioda,thatareparalleltothesubstrate.
surrounding the spherical metallic particles. While the
The donor array is placed at the height h from the sub-
general approach is similar, the electromagnetic modes
strate, the acceptor array lies in the plane z =h+δh at
for the spherical geometry are quite different, so the re-
the vertical distance δh from the acceptor array. The x
sults of Ref. 18 cannot be applied to our system directly.
and y axes are parallel to the interface plane z =0.
Hence, there is a growing demand from both theory and
experiment to elaborate on the energy transfer in planar Weusethesemiclassicalcoupleddipoleapproximation
plasmonic systems with large molecule concentration. with the random source terms, similarly to the approach
2
in Refs. 18,19: rate Γ of the excited acceptor state is assumed enough
A
fast to quench both the acceptor-acceptor coupling and
1 (cid:88)
p =f + G(r −r )p , (1) possible transfer of the energy back to the donors.
αD D,j D,j D,j D,j(cid:48) D,j(cid:48) Wewillfirstcalculatethepopulationofdonorsbysolv-
j(cid:48)(cid:54)=j
ing Eq. (1). This can be accomplished by expanding the
1 (cid:88)
α pA,j = G(rA,j −rD,j(cid:48))pD,j(cid:48) . (2) donor dipole momenta over the Bloch modes, character-
A j(cid:48) ized by the in-plane Bloch wave vector k:
(3)
(cid:88)
pD,j = eikrD,kpD,k
Here p (p ) are the dipole momenta of the donors
D,j A,j k
(acceptors) located at the points rD,j (rA,j) and having a2 (cid:90)(cid:90)
thepolarizabilitiesαD (αA), G(r,r(cid:48))isthetensorGreen ≡ (2π)2 dkxdkyeikrD,kpD,k. (8)
function
|kx,y|<π/a
(cid:16)ω(cid:17)2 (cid:16)ω(cid:17)2
∇×∇×G(r,r(cid:48))= ε(r)G(r,r(cid:48))+4π ˆ1δ(r−r(cid:48)), Substituting Eq. (8) into Eq. (1) we obtain a system of
c c independent equations for the Bloch amplitudes p .
(4) D,k
Solving them and substituting the result back to Eq. (8)
accounting for the presence of the metallic mirror,20 and
we obtain
f aretherandomsourceterms. Weassumeincoherent
D,j
(cid:88)(cid:88)
stationary pumping of the donor array, in this case the pD,j = eik(rD,j−rD,j(cid:48))α˜D,kfD,j(cid:48) . (9)
realization-averagedvaluesoff areequaltozero. The
D,j k j(cid:48)
pumping to different molecules can be considered inde-
where
pendent and is characterized by the following correlator:
α
α˜ = D , (10)
(cid:104)fD,j,µfD,j(cid:48),ν(cid:105)= 2Sπδµνδjj(cid:48)δ(ω−ω(cid:48)), (5) D,k 1−αDCD,k
and
here S is the effective rate of the generation of excitons
(cid:88)
in donors19, µ,ν = x,y,z and the brackets denote the CD,k = e−ikrD,jG(−rD,j) (11)
averaging over the realizations of the random sources. j(cid:54)=0
The polarizabilities have the resonant form,
is the interaction constant for the planar donor array.22
d2 Eq. (9) fully accounts for the hybridization between the
α = M , M =D,A, (6)
M (cid:126)[ω −ω−i(Γ +Γ )] donorarrayandtheelectromagneticmodesofthemetal-
M M 0,M
lic substrate, including the plasmonic modes. This is
where d are the matrix elements of the dipole moment encoded in the polarizabilities of the Bloch modes α˜ ,
M D,k
for donors and acceptors, ω are the resonant frequen- which have the form of the bare donor polarizability α
M D
cies and Γ are the phenomenological nonradiative decay renormalized by the interaction with the mirror. The in-
rates. The radiative decay rate Γ is then described by teraction constant C can be evaluated either by Flo-
0 D,k
the standard expression21 quer summation22 or by an Ewald-type summation23,
√ here we resort to the latter procedure.
Γ = 2ωM3 d2M εb . (7) Theaverageoccupationnumberofthedonormolecules
0,M 3(cid:126)c3 can be introduced as the averaged squared amplitude of
the donor dipole momentum divided by the dipole mo-
Ourgoalistodeterminethestationarypopulationofac-
mentum matrix element:
ceptors induced as a result of the pumping of the donors
and the energy transfer from donors to acceptors. Im- 1
N ≡ (cid:104)|p (t)|2(cid:105)
portantly, inEq.(1)weincludetheelectromagneticcou- D |d |2 D
D
pling between the donor molecules (second term in the
right hand side). This coupling leads to the formation = 1 (cid:90)(cid:90) dωdω(cid:48)ei(ω−ω(cid:48))t(cid:10)p∗ (ω)·p (ω(cid:48))(cid:11) .
|d |2 (2π)2 D,j D,j
of the collective states, eigenmodes of Eq. (1). Impor- D
(12)
tantly, the collectiveeffectsaremanifestedevenforinco-
herent pumping. The only necessary conditions are the
Substituting Eq. (9) into Eq. (12) and using the random
weak inhomogeneous broadening in the donor array and
source correlation function Eq. (5) we obtain
longenoughlifetimeofthedonorstates. Inthelinear-in-
pumping regime the pumping determines only the pop- (cid:90) dω (cid:88)
N =S Trα˜ α˜† . (13)
ulation of the different eigenmodes, but not their spatial D 2π D,k D,k
structure. In order to simplify the model we neglect the k
electromagnetic coupling between acceptors in Eq. (2). Now we will assume weak coupling between the donor
The reason is that the energy is transferred to the ex- array and the mirror. In this approximation the interac-
cited states of the acceptors. The nonradiative decay tionwiththemirrorleadsonlytothemodificationofthe
3
donor lifetime. The complex polarizabilities α˜ then is the Green function Fourier component describing the
D,k
have the form donor-acceptor coupling. In the chosen approximation
the lifetime of the (excited) acceptor states is deter-
d2
α˜D,k = (cid:126)[ω −ω−i(ΓD +F Γ )] , (14) mined solely by energy relaxation processes, ΓA (cid:29)
M D k 0,D Γ ,Γ ,Γ ,1/τ . Hence,wecanintroducetheeffec-
0,A 0,D D k,λ
tiveenergytransferrate1/τ fromthefollowingkinetic
where ET
equation for the acceptors:
3
Fk =1+ 2(ωD/c)3√εb ImCk(ωD), (15) NA = ND . (22)
τ τ
A ET
is the tensor analogue of the Purcell factor, describing
the lifetime modification. The matrix F can be diago- The left-hand side of this equation describes the (non-
k
nalized, radiative) decay of acceptors with the lifetime τA =
1/(2Γ ),theright-hand-sidedescribesthetransferofen-
A
F =(cid:88)3 V f [V−1] , µ,ν =x,y,z, (16) ergy from donors with the population ND. Integrating
k,µν k,µλ k,λ k λν over frequency in Eq. (20), using the donor population
λ=1 N fromEq.(17)andthetransferratedefinitionEq.(22)
D
we obtain
where f are the Purcell factors characterizing the
k,γ
decay of the Bloch modes with different polarizations. 1 2π 1 Γ
There exist three possible polarizations for each value of τ = (cid:126) π(ω −ωA)2+Γ2 |dA|2|dD|2G2, (23)
k. Note,thatthematrixV inEq.(16)isingeneralcase ET D A A
k
neither Hermitian nor symmetric. After the frequency where the effective constant of the transfer rate reads
integration in Eq. (13) is carried out, the result for the
donor population assumes the form G2 = W ≡ 1 (cid:88)τ |G |2 (24)
T (cid:80)τ k,λ DA,k,λ
(cid:88) k,λ k,λ
N =S τ (17) k,λ
D k,λ
k,λ with
where
G =e [G ] V . (25)
DA,k,λ µ DA,k µν k,νλ
1
τ = (18)
k,λ 2(Γ +f V∗ [V−1] ) Equations(23)–(25)fortheeffectiveenergytransferrate
D k,λ k,λµ k µλ
constitute the central result of this study. Their struc-
is the effective lifetime of the Bloch mode and the sum- ture is quite clear. The first resonant factor in Eq. (23)
mation over the dummy index µ is assumed. Eq. (17) depends on the spectral detuning between donor and ac-
hasaclearphysicalmeaning: thetotalpopulationofthe ceptor molecules. The second factor G2 describes the
donors is a sum over the populations of different eigen- electromagnetic coupling between the molecules arrays.
modes. Eacheigenmodepopulationisgivenbytheprod- Ifthecollectivecouplingbetweenthedonorsisneglected,
uct of the pumping rate and the lifetime. Since donors the factor reduces to
are pumped in an uncorrelated way, the pumping con-
(cid:88)
stantisindependentoftheBlochvectork. However,the G02 = Tr[G†(rA,j −rD,j(cid:48))G(rA,j −rD,j(cid:48))], (26)
lifetimes τ can differ for different modes. j(cid:48)
k,λ
Nowweproceedtothecalculationoftheacceptorpop-
i.e. it is described by a sum of individual transfer events
ulation. To this end we substitute Eq. (9) into Eq. (2)
from different donors. In this case the transfer rate as-
and calculate the correlation function
sumes the same form as in Refs. 14–16. In the general
1 case,however,thetransferconstantG2 isgivenbyasum
N ≡ (cid:104)|p (t)|2(cid:105)
A |d |2 A of the contributions from Bloch eigenmodes of the donor
A
ensemble. Sincethedonorsarepumpedincoherentlyand
= 1 (cid:90)(cid:90) dωdω(cid:48)ei(ω−ω(cid:48))t(cid:10)p∗ (ω)·p (ω(cid:48))(cid:11) . (19) uncorrelated, the contributions are independent. Each
|d |2 (2π)2 A,j A,j
A term in the sum in Eq. (23) is proportional to the life-
timeofthecorrespondingmodeτ ,thatdeterminesthe
This yields (cf. with Eq. (13) and see also Ref. 24) k,λ
mode population.
(cid:90) dω (cid:88)
N =S |α |2 TrG α α† G† (20)
A 2π A DA,k D,k D,k DA,k
k III. RESULTS AND DISCUSSION
where
In this section we present detailed analysis of the ef-
(cid:88)
GDA,k = eikrA,jG(rA,j −rD,j(cid:48)) (21) fect of the metallic mirror on the transfer rate Eq. (24).
Figure2showsthedependenceofthetransferrateonthe
j
4

(a) out-of-plane transfer
0.4 x0.3
0.3
h = 10 nm
0.2
arb.un) 0.1 103 h = 20 nm
e ( (b) in-plane transfer
Transfer rat 12..50 x1.4 cell factor 101
ur
x1.4 a = 7 nm, collective P
a = 15 nm, collective
1.0 a = 7 nm, single donor
a = 15 nm, single donor -1
10
5 10 15 20 25 30
height above the mirror (nm)
FIG. 2: (Color online) Energy transfer rate as function of 0.01 0.1 1
the height of the donors above the mirror h for (a) acceptors
Bloch wave vector ( /a)
located above donors and (b) acceptors located in the same
horizontal plane as donors. Thick solid/red and dashed/blue FIG.3: PurcellfactorsfortheBlochmodesofthedonorar-
curvescorrespondtothetransferratescalculatedaccordingto raywithdifferentpolarizationsasfunctionsoftheBlochwave
Eq. (24) with the lattice constants a=7 nm and a=15 nm, vector along the Γ–X direction. Thick/black and thin/red
respectively. The curves are normalized to the transfer rates curvescorrespondtoh=10nmandh=20nm,respectively.
betweensingledonorandsingleacceptoratthecorresponding Calculatedfora=7nmandthesameotherparametersasin
distance without the mirror. Thin curves show the transfer Fig. 2. The arrow indicates the value of k equal to the light
√
ratesbetweensingledonorandsingleacceptor,i.e. calculated cone edge ω ε /c.
D b
fromEq.(26)neglectingthecollectiveeffects. Thecalculation
parameters are as follows: λ = 2πc/ω = 500 nm, ε = 2,
D b
Γ = 0.1Γ , ε = −9.77+0.31i (corresponds to Ag in
D 0,D subs donors, the transfer is strongly suppressed by the mir-
Ref. 25), r −r = ∆hzˆ, ∆h = 3.5 nm (a) and r −
A,j D,j A,j ror, while for acceptors located between the donors the
r =a/4(xˆ+yˆ)(b). Theinsetsschematicallyillustratethe
D,j transfer is enhanced. This effect cannot be described in
geometry of the problem.
theindependenttransfermodel(thincurves),whichpre-
dictsalmostvanishingdependenceofthetransferrateon
the distance up to the distance h ∼ 3 nm, in agreement
heightofthedonorandacceptormoleculesabovethemir- with Refs. 12,13. The transfer modification is most pro-
ror. Thelatticeconstantofthedonorandacceptorarrays nounced when the height of the donor array above the
andtheirrelativepositionswithrespecttoeachotherre- mirror is on the order of the array period a (cf. solid
main fixed. Two panels of Fig. 2 correspond to different and dotted curves for a=7 nm and a=15 nm). As the
geometries: (a)out-of-planetransfer,whenacceptorsare period increases, the effect is suppressed. Thus, collec-
positioned above the donors at the height δh and (b) in- tive transfer model predicts a strong modification of the
planetransfer,whenacceptorsarepositionedinthesame transfer rate near the metallic mirror, in stark contrast
horizontal plane as the donors so that each acceptor lies to the independent transfer model. Now we proceed to
in the center of the square unit cell of the donor lattice. the explanation of these findings.
ThicklinesarecalculatedaccordingtoEq.(25)andfully Ouranalysisindicates,thattheGreenfunctions,enter-
account for the hybridization between the donors while ing Eq. (25), are not very sensitive to the distance from
thin lines are calculated in the independent donor ap- the mirror. Hence, the strong transfer modification, ob-
proximation Eq. (26) by retaining only one term for one served in Fig. 2 is solely due to the dependence of the
donor, that is closest to the acceptor. Importantly, the lifetimes of the Bloch eigenmodes τ on the distance.
k,λ
transfer rates in the collective model and the indepen- ThePurcellfactorsf ,characterizingthelifetimemod-
k,λ
dent donor model are different from each other even in ification by the mirror, are presented in Fig. 3. Thick
the absence of the mirror (or for h (cid:29) a). The reason is andthincurveswerecalculatedasfunctionsoftheBloch
that the local field of the donor array at the acceptor is wavevectorfortwodistances,h=10nmandh=20nm,
differentfromthatofthesingledonoreveninvacuum. In respectively. Foreachheightweshowthreecurves,corre-
order to compare the two models visually, the curves for sponding to different eigenmode polarizations. The cal-
the single-donor model have been scaled by the factors culation reveals strong dependence of the Purcell factors
0.3and1.4inFig.2aandFig.2b,respectively. Themain on the wave vector, distance and polarization. For given
resultofthiswork,showninFig.2,isasfollows: thecol- distance one can distinguish between several regions in
lective transfer rate is strongly sensitive to the distance Fig.3. Thefirstregionisinsidethelightcone(k <ω/c).
fromthemirror(thicklines)whiletheindependent-donor In this case the Purcell factor is much larger than unity
one is not. When the acceptors are positioned above the even at the large distances from the mirror (or without
5
the mirror). This can be understood as a semiclassi- modes on the Bloch wave vector for out-of-plane (black
cal counterpart of the Dicke superradiance,26 when the solid curves) and in-plane transfer (red dotted curves).
donors, that are separated by the distances smaller than For each geometry we have plotted the Green function
the light wave length, emit collectively. The radiative component only for one polarization, that is largest by
rate is larger by the factor on the order of 1/(ωa/c)2 the absolute value and provides the dominant contribu-
than that of the individual donor in vacuum.27 Second, tion to the transfer rate. The Green function values in
one can see the peak in Fig. 3 at k ∼ 0.04π/a, that de- panel (a) have been additionally multiplied by the fac-
scribesstrongenhancementofthedonordecayduetothe tor k, accounting for the growth of the density of states
plasmons. Thepeakispresentonlyforoneofthecurves, d2k ≡ kdkdϕ with the absolute value of the wave vec-
k
correspondingtothetransversemagnetic(TM,orp)po- tor. The transfer rate Eq. (24) is proportional to the
larization. Third, there exists a region of larger Bloch integral of the product of the values of k|G |2 in
DA,k,λ
wave vectors k (cid:38)0.1π/a. This range corresponds to the Fig. 4(a) and τ in Fig. 4(b) over the Bloch wave vec-
k,λ
evanescent modes, with the electric field exponentially tor k. Importantly, due to the difference in the Green
decayingfromthedonorplaneandlyingoutsidethelight function values |G |2, the out-of-plane transfer is
DA,k,λ
cone. In the absence of the mirror these waves would be mediated by the p-polarized Bloch modes, while the in-
completelyevanescentandtheywouldhavezeroradiative plane transfer is controlled by the s-like modes. These
decay rate, f = 0. Due to the presence of the mirror, modes have quite different lifetimes, that are plotted in
k
the modes acquire finite lifetime due to the absorption Fig. 4b for the height h = 20 nm above the mirror (see
in the substrate, f ∝ Imε e−2kh.1 The weakest val- also Fig. 2). This explains the qualitative difference be-
k subs
ues of the Purcell factor in this third region correspond tween the dependence of the in-plane transfer and the
to the donor modes with the polarization close to trans- out-of-plane transfer on the distance from the mirror. In
verse electric (TE, or s). Although the Purcell factor ordertoelucidatetheeffectwewillnowanalyzethesetwo
for the evanescent modes is much less than unity, it is casesinmoredetail. (i)Out-of-planetransfer. Whenthe
stronglyenhancedwhenthearrayapproachesthemirror array approaches the mirror, the lifetime of the p-type
and the coupling to the mirror increases (cf. black and modes, determining the transfer, becomes shorter. The
red curves). We will now explain how this very effect product W = (cid:80) τ |G |2, controlled by the p-
k,λ k,λ DA,k,λ
leads to the modification of the transfer rate Eq. (24) by type modes, is decreased. On the other hand, the total
(cid:80)
the mirror. lifetime T = τ of the donor array is due to the
k,λ k,λ
Figure 4 shows the dependence of the Green func- long-living modes with s-type polarization, rather than
tion components Eq. (25) and the lifetimes of donor the p modes. The lifetime of s-type modes is equal to
1/2Γ for k (cid:38)0.1π/a and weakly sensitive to the mirror
D
(red/dotted curve in Fig. 4b). As a result, W decreases

faster than the T when the mirror is brought closer, and
300
(a) thetransferrateEq.(24)∝W/T issuppressed,inagree-
out-of-plane transfer ment with Fig. 2(a). (ii) In-plane transfer. In this case
7 a) 200 in-plane transfer the function W is determined by the long-living s-type
2 | (A,k, eigenmode. Hence, the value of W is practically insensi-
D tivetothemirrorbecausethelifetimeismodifiedonlyfor
G 100
k| very small k (cid:46)0.1π/a, where the product of the squared
Greenfunctionandthedensityofstatesissmall. Hence,
0 W decreasesslowerthanT,andthetransferrateW/T is
10 (b) enhanced at smaller distances from the mirror, in agree-
ment with Fig. 2(b).
)D 8
0,
6

(
4
k,
2
0
Finally, in Fig. 5 we analyze the dependence of the
Bloch wave vector
transfer rates on the nonradiative decay of the donors
FIG. 4: Green function Eq. (25) characterizing the transfer
Γ . Clearly, when the nonradiative component of the
strength (a) and lifetimes of the Bloch modes of the donor D
decay rate increases, the decay rate of any given Bloch
array (b) as functions of the Bloch wave vector in the Γ–
M direction [inset in panel (b)]. Black/solid and red/dotted mode becomes less sensitive to the mirror presence. As
curvescorrespondtotheout-of-planeenergytransferandin- aresult, thetotalenergytransferratereturnstoitsbulk
planetransfer,asindicatedbytheinsetsinpanel(a). Foreach value,unaffectedbythemirror. Thisisinagreementwith
geometryonlythemodewithonepolarization,providingthe the result of calculation in Fig. 5: both for out-of-plane
largest contribution to the energy transfer, is shown. Other (red/dottedcurve)andin-plane(black/solid)geometries
calculation parameters as the same as in Fig. 2. the energy transfer tends to its bulk value for large Γ .
D
6

rayscanbestronglymodifiedbythemirror,instarkcon-
n.) 1.2 out-of-plane transfer trast to the transfer between individual molecules. The
nsfer rate (arb. u 11..01 in-plane transfer emcononilhrlterahocnetrciverisdeeldasBetuilnevosecithtopivomtishtioyetdioomefnsothbdoefeifitttwcrhaaeetneinsodfnoetrnhooetfordtthaohreneroalpriyfr.eaetnsiDmedneeacpsececooneffdpttitnhhogeer
Tra 0.9 molecules, the transfer can be both suppressed (accep-
tors between the donors) or enhanced (acceptors above
0.8
1 2 3 4 the donors). While further study is needed to account
Nonradiative decay rate D ( 0,D) for the possible effect of the strong coupling between the
donors and the plasmons,28 for inevitable effects of dis-
FIG. 5: (Color online) Energy transfer rate as function of
the nonradiative decay of the single donor Γ. Black/solid order in actual samples, and for the bulk film geometry,
and red/dotted curves correspond to the out-of-plane energy these our results might already provide the key to ex-
transfer and in-plane transfer, as indicated by the insets. plain the suppression of the energy transfer in the plas-
Other calculation parameters as the same as in Fig. 2. monic environment, recently observed experimentally in
Ref. 13.
IV. SUMMARY
Acknowledgements
To summarize, we have developed a theory of energy
transferbetweentheplanararraysofdonorandacceptor
molecules,lyingontopofametallicmirrorintheregime I acknowledge useful discussions with M.A. Nogi-
of weak, incoherent and spatially uncorrelated pumping nov, A.V. Rodina, E.L. Ivchenko, E.E. Narimanov,
ofthedonors. Wehaveusedacoupleddipolemodelwith T.V. Shahbazyan, and P. Ginzburg. This work has
random source terms that has allowed us to obtain ana- been funded by the Government of the Russian Feder-
lytical results for the transfer rate. It has been demon- ation (Grant No. 074-U01), “Dynasty” Foundation and
strated, that the transfer rate between the molecule ar- RFBR.
∗ Electronic address: [email protected]ffe.ru 203601 (2012).
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