Coherent light scattering and resonant energy transfer in an apertureless scanning near-field optical microscope

PHYSICAL REVIEW B, VOLUME 63, 045420

Coherent light scattering and resonant energy transfer in an apertureless scanningnear-field optical microscope

Jaromı´r Fiurasek, Boris Chernobrod, Yehiam Prior, and Ilya Sh. Averbukh*

Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel~Received 18 June 2000; revised manuscript received 23 October 2000; published 9 January 2001!

We investigate the interaction of two molecules or nanosized particles with a nearly resonant laser fieldunder the tip of an apertureless near-field microscope. We show that interference of several scattering channelsprovides means for enhanced spatial resolution. The visibility of two separate nano objects is considered, anda natural definition emerges for the resolution of the apertureless microscope operating under conditions ofnearly resonant illumination. The probe tip creates an additional coupling channel between the two molecules,and thus affects the energy transfer between them. We demonstrate that the tip can either enhance or suppressthis transfer. Two models for the tip geometry are considered: a simplified pointlike dipole, and a more realisticelongated spheroid. Quantitative results are obtained for the dependence on irradiation frequency and tipposition for dielectric as well as metallic tips. In particular, specific results are obtained for a silver tip underconditions of plasmon resonance, and we show that under fully resonant conditions the tip may enhance theintermolecular energy transfer by nearly two orders of magnitude.

DOI: 10.1103/PhysRevB.63.045420 PACS number~s!: 87.64.Xx, 41.20.Cv

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I. INTRODUCTION

Recent progress in the spectroscopy of single moleculeclosely related to advances in microscopy at suboptwavelength resolution obtained via scanning near-field ocal microscopy~SNOM!.1–8 Many variants of near-field optical microscopes employ single-mode optical fibers tapeat their end and~possibly! metal coated to form a subwavelength aperture. The typical resolution of these systems ithe order of tens of nanometers, limited by the penetrationevanescent fields through the metal coating of the fibMoreover, aperture-based scanning near-field optical miscopes often suffer from limitations on the optical powwhich can be delivered through the subwavelength apertBoth of these problems are alleviated by aperturelschemes,9–19 in which a sharp nanosized probe near the sface is illuminated from the outside by an external ligsource. This approach is based on an enhancement of opfields in close proximity to a sharp tip. The enhancemeffect has a twofold origin: First, the field increases due tpure geometrical reason, the so-called ‘‘lightening roeffect,20–22 and further enhancement arises due to excitaof localized plasmons in a metallized tip.13,20,23

Localized strong fields near the sharp tip enable varinanoscopic applications. The field enhancement locallycreases the efficiency of nonlinear optical processes suctwo-photon absorption24,25 or local second-harmonic genertion from a rough metallic surface.26 Resonantly excited surface plasmons propagating along a silver/air interface calocally probed by an apertureless SNOM tip.27 A small di-electric particle, lying in a vicinity of the sharp tip, is subjected to strong forces arising from large near-field gradieso that the tip can serve as nanometric optical tweezerslowing for trapping and alignment of dielectrinanoparticles.28

In scanning experiments, near-field interactions betwthe probe and the sample can dramatically enhance

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scattering from the observed object, because the tip actan efficient ‘‘antenna,’’ and the tip-object system scattelight in a cooperative manner. As a result, such a systemthe potential of achieving single-molecule sensitivity, espcially when exciting resonant molecular transitions.14,29,30

Localization of the enhanced near field in the vicinitythe tip enables one to examine local properties of the samAs we recently showed, coherent tip-sample interactiomanifest themselves in the dependence of the total scattintensity on the amplitudes and phases of the complex poizabilities of the sample and the tip.30 As a result, sub-nanometer localization of a single object~e.g., a molecule! ispossible when the scattered intensity is detected. Howeone has to distinguish between localization and resolutand in order to define the resolution of such experiments,detection of at least two nearby objects must be conside

In this paper, we examine the detection of two closelying molecules~or other small objects! by an aperturelessSNOM device. Molecules are described as pointlike dipowith well defined resonant frequencies, polarizabilities, adecay rates. In particular, we concentrate on the case ofwhich is nearly resonant with the molecular transitions, athe tip plasmon resonances are also not too far detuWhen the distance between the molecules is small enothey can exchange energy via resonant dipole-dipole intetion even in the absence of the tip. An approaching tip, hoever, significantly modifies the molecular attributes~resonantfrequencies and decay rates!,31–36 and we show that, in addition, the tip creates another coupling channel betweenmolecules. As a result, a properly positioned tip can be uto control the energy transfer, or more generally the cpling, between the molecules. A discussion of the eneexchange between two atoms, one inside a dielectric misphere and the other placed near its outer surface, wacently presented.36

Theoretical descriptions of the tip-sample interactionan apertureless SNOM device were based on numericalutions of Maxwell equations, and several numeric

©2001 The American Physical Society20-1

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FIURASEK, CHERNOBROD, PRIOR, AND AVERBUKH PHYSICAL REVIEW B63 045420

schemes were suggested, such as the multiple-multimethod34,28 and the Green dyadic technique33,37,38~for a re-view, see Girard and Dereux39!. These numerical simulationprovide the tools to investigate realistic tip-sample geoetries. However, in order to gain insight into the physicsthe image formation or the intermolecular energy transfeis useful to employ simple geometrical models for the tallowing for analytical description. In this paper, we use twdifferent models of the tip. We begin with a simple descrtion; the tip is represented by a pointlike dipole, which eables us to illustrate the main features of the resonant lscattering and energy transfer. To account for the finite sof the probe, we then consider a spheroidal tip. The papeorganized as follows: In Sec. II, the resonant light scatterfrom the tip-sample system is studied utilizing the pointlidipole model for the tip. In Sec. III, the same model of ttip is used to investigate the tip-induced modifications ofenergy transfer between two molecules. The spheroidal tconsidered in Sec. IV, and the conclusions are presenteSec. V.

II. STATIONARY SCATTERING MEDIATED BY APOINTLIKE DIPOLE TIP

In this section we consider the simplest case of two mecules~dipoles! on the surface, interacting with a pointlikdipole tip hovering over them. The geometry of the probleis the following: two molecules lie onx axis at pointsx56d, and the probe tip moves over them at a fixed vertiheighth. We assume for simplicity that the tip and molecudipoles and the external field are all oriented alongz axis.We study a stationary response of the tip-molecule sysilluminated by strong coherent laser excitation. The distanbetween the molecules and the probe are assumed tothe range of a few nanometers to tens of nanometersthese distances the retardation effects and the change ocitation fieldE0 can be neglected, and one can use an etrostatic description. We denote the polarizabilities of tmolecules and the tip bya1 , a2, andaP , respectively. Thedipole moment of each dipole is given by a product ofpolarizability and total electric field at the position of thdipole. This field consists of two components, the first isexternally applied excitation fieldE0, and the second is fieldgenerated by the other dipoles. This generated field cancalculated with the use of the so-called Green dyadic fution G,

G~r ,r 8!53n^ n2 I

ur2r 8u3, ~1!

wheren5(r2r 8)/ur2r 8u and ^ denotes the tensor produof two vectors. In the steady state, the amplitudes of dipmomentsm j can be found as a solution of the following sof self-consistent coupled linear equations:

m15a1~E01G12m21G1PmP!,

m25a2~E01G21m11G2PmP!, ~2!

04542

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mP5aP~E01GP1m11GP2m2!.

Indices 1, 2, andP correspond to the first and second moecules, and to the probe, respectively. Coupling constantsgiven by

Gi j 5Gzz~r i ,r j !, i , j 51,2,P. ~3!

A solution of the Eqs.~2! yields the dipole moments. Wedefine the total dipole moment of the system as

m tot5m11m21mP . ~4!

This total dipole moment is proportional to the excitatiofield, the proportionality constant being the total polarizabity of the system:

a tot5m tot /E0. ~5!

The exact expression fora tot can be found by direct solutionof system~2!. This expression has a complicated form thcan be considerably simplified when the field of thegreatly exceeds the external fieldE0 and the field of themolecular dipoles. Such strong local-field enhancemenpossible with a metallic tip near plasmon resonance,30 whosepolarizability aP is typically much larger than resonant optical polarizabilities of moleculesa j . Of course, the tip musbe placed close enough to the molecules so thatr jP,aP

1/3

holds.Under the above conditions we can keep only domin

termGjPmP in the first two of Eqs.~2!, and insert the result-ing expressions form1 andm2 into the third equation. Thuswe obtain

a tot'aP

12aP~a1G1P2 1a2G2P

2 !. ~6!

The total polarizability is sensitive not only to the amplitudbut also to the phases ofa j . We recall that the phase ofa jchanges from 0 top when one scans the excitation frequency through the relevant resonance line. For a molecdipole characterized by a resonant frequencyv j and decayrateg j we can write

a j5a j ,res

D j2 i, ~7!

whereD j5(v2v j )/g j is the normalized molecular detuning. Near its plasmon resonance, the tip polarizabilityaPbehaves in a similar way. The whole system radiates likpointlike dipole with a dipole momentm tot . The intensity ofscattered light is thus proportional to the square ofm tot :

I scatt}um totu25ua totu2E02 . ~8!

Here I scatt represents the signal which is detected in the ffield zone. The strength of the signal measured by the detor significantly depends on the angle of observation. In pticular, the substrate which holds the molecules may stronmodify the pattern of the radiating dipole.40 In the experi-ment, it would be optimal to place the detector in the diretion of the maximum dipole radiation in order to achieve t

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COHERENT LIGHT SCATTERING AND RESONANT . . . PHYSICAL REVIEW B 63 045420

highest possible signal-to-noise ratio. Therelative modula-tion of the near-field signal studied in this paper, howevdoes not depend on the position of the detector.

Figures 1–4 demonstrate the scanning signals from amoving above two molecules. All figures were obtainedexactly solving system of equations~2! and calculatingm totaccording to Eq.~4!. We see that the signal can be eithenhanced or suppressed depending on the phases of molar and tip polarizabilities. Figures 1 and 2 show the scatteintensity for an off-resonant tip (aP is real!. In this case, themolecules can be most clearly resolved when their resofrequencies are slightly different and the laser is tuned hway between their resonances. A three-dimensional plothe signal for such a case is shown in Fig. 2. Note that e

FIG. 1. Normalized scattered intensity from the tip-molecucompound for a tip scanning over thex axis at a fixed verticalheight h510 nm. The stars denote the molecular positionsx565 nm). The resonant molecular polarizabilities area1,res

5a2,res53 nm3, and the probe has a polarizabilityaP5103 nm3.The scanning signal is shown for four different normalized molelar detuningsD j5(v2v j )/g j : D15D250 ~solid line!, D15D2

51 ~long dashed line!, D15D2521 ~dot dashed line!, and D1

50.75 andD2520.75 ~short dashed line!.

FIG. 2. Normalized scattered intensity for a tip scanning othe (x,y) plane at fixed vertical heighth510 nm above two mol-ecules placed on thex axis atx565 nm. aP51000 nm3, a1,res

5a2,res53 nm3, and the normalized molecular detunings areD1

51 andD2521.

04542

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two molecules of the same species can have slightly differesonant frequencies if their local environments are not copletely identical. If the molecular resonances coincide,resolution strongly decreases and the two molecules behalmost like a single object. In particular, when both moecules are in resonance, the signal exhibits almost no vation.

Near the plasmon resonance, a metallic tip is characized by large purely imaginary polarizabilityaP . The scan-ning signals shown in Figs. 3 and 4 were obtained foresonant tip polarizability ten times larger than the oresonant one employed in Figs. 1 and 2. Due to the plasmresonance, the modulation of the scattered intensitystrongly enhanced in comparison to the probing by the oresonant tip, and the two molecules can be clearly resolvThe behavior of the scanning signals can be explained wthe help of the approximate formula@Eq. ~6!#. For a purelyimaginary aP , the interference between direct scatteri

r

-

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FIG. 3. Normalized scattered intensity for a resonant tip (aP

5 i 3104 nm3) scanning over thex axis at a fixed vertical heighh510 nm above two molecules placed on thex axis at65 nm.The molecular parameters area1,res5a2,res53 nm3 and D15D2

50 ~solid line!, D15D251 ~dashed line!, andD15D2521 ~dotdashed line!.

FIG. 4. Normalized scattered intensity for a resonant tip scning over the (x,y) plane.D15D250, and all other parameters arthe same as in Fig. 3.

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from the tip and tip-mediated molecular scattering is alwadestructive, irrespective of the molecular detunings. As asult the scanning signal is suppressed, as shown in Figand 4. The scanning signal can be enhanced only if thenominator modulus in Eq.~6! is smaller than unity. Thisconstructive interference can occur only if the real partsaP anda1 ~or a2) are dominant and have the same sign;Figs. 1 and 2.

III. RESONANT ENERGY TRANSFER

To study the energy transfer between two moleculesthe presence of the probe, one first has to consider theporal evolution of the system in the absence of an extefield. Generally, we can treat both molecules and theequivalently, introducingvP , gP , andaP,res, and we haveto deal with system of three coupled dipoles.41 Assumingthat all the frequencies do not differ too much from eaother, we can use the rotating-wave approximation, introding variables

m j~ t !5pj~ t !e2 ivt, ~9!

wherev is a reference frequency~e.g.,v1). The dynamics ofthe system is described by the following equations:

S d

dt1g11 i ~v2v1! D p15 ig1a1,res~G12p21G1PpP!,

S d

dt1g21 i ~v2v2! D p25 ig2a2,res~G21p11G2PpP!,

~10!

S d

dt1gP1 i ~v2vP! D pP5 igPaP,res~GP1p11GP2p2!.

In our case the probe tip is a dielectric or metallic bodwhose relaxation time is much shorter than the lifetimethe molecules. Thus we can assume that the dipole momof the probe follows adiabatically the local field, and we creplacepP with

pP5aP~GP1p11GP2p2!, ~11!

whereaP is given by Eq.~7!.With the help of Eqs.~10!, ~11!, one obtains

S d

dt1g11 i ~v2v1!2 ig1a1,resaPG1P

2 D p1

5 ig1a1,res~G121aPG1PGP2!p2 ,

S d

dt1g21 i ~v2v2!2 ig2a2,resaPG2P

2 D p2

5 ig2a2,res~G211aPG2PGP1!p1 . ~12!

Several physical phenomena can be seen from the struof Eqs. ~12!. As follows from the left-hand sides of thesequations, the presence of the tip leads to a renormalizaof the resonant frequencies and the decay rates of theecules:

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G15g1@11a1,resIm~aPG1P2 !#,

G25g2@11a2,resIm~aPG2P2 !#,

~13!v15v11g1a1,resRe~aPG1P

2 !

v25v21g2a2,resRe~aPG2P2 !.

A coupling between these perturbed molecules is characized by a~complex! coupling constant on the right-hand sidof Eqs.~12!:

C5G121aPG1PGP25G211aPG2PGP1 . ~14!

We can see that there are two coupling channels. The dcoupling is proportional toG12, and the coupling mediatedby the tip is proportional toaPG1PGP2. The distancer 0 atwhich these two channels become comparable can bemated as

r 0'~aP!1/3. ~15!

We assume that initially only one molecule is excited, awe shall determine the total energy which is transferredthe second molecule. The driving force acting on the sec~initially unexcited! molecule is given by

F~ t !5eCp1e2 ivt1c.c., ~16!

wheree is the dipole charge. The energy transfer rateR(t) isa product of this force and the velocity of the second dipoAveraging over the fast optical oscillations, and taking inaccount thatu ivp2u@u p2u, we can write

R~ t !52 iv~p2~ t !C* p1* ~ t !2c.c.!. ~17!

Thus we obtain the following result for the total energtransferred from the first to the second dipole:

W5E0

`

R~ t !dt52v ImS C* E0

`

p2~ t !p1* ~ t !dtD . ~18!

The solution of the coupled differential equations~12! for theinitial condition p15p1(0) andp250 can be easily derivedby standard means. If the coupling between the two mecules is weak and only a small part of the energy is traferred to the second molecule, then the back action ofsecond molecular dipole on the first one can be neglectedthis paper we shall analyze this weak-coupling regime.principle, however, the intermolecular interaction maystrong, which may lead to a significant modification of tscattered light spectrum41 and a strong back action influencing the resonant energy transfer.

Treating the coupling as a small perturbation, we cwrite

p1~ t !5p1~0!exp@2G1t2 i ~v2v1!t# ~19!

and

p2~ t !5 ig2a2,resCE0

t

e2[G21 i (v2v2)]( t2t)p1~t!dt. ~20!

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This yields

p2~ t !5 ig2a2,resCe2G1t1 i v1t2e2G2t1 i v2t

~G22G1!1 i ~v12v2!e2 ivtp1~0!.

~21!

Inserting Eqs.~19! and ~21! into Eq. ~18! we derive an ex-pression for the total transferred energy

W'va2,resg2S 11G2

G1D uCu2

~G11G2!21D2up1~0!u2. ~22!

Here D5v12v2 is the tip-modified detuning of the molecules.

The derived analytical results and the numerical calcutions reveal several important features of the system.energy transfer between two molecules is substantially inenced by the presence of the probe. There are two enetransfer channels, a direct channel and a probe medichannel, and these two channels can interfere so thatenergy transfer can be either enhanced or suppressedprobe tip also modifies the molecular decay ratesG j andresonant frequenciesv j . The higher the decay rate, thlower the energy transfer~the molecules have less timeexchange energy!. In order to characterize the influencethe tip on the mutual interaction of the molecules, we usenormalized ratiosW/W0, whereW0 is the energy transfer inthe absence of the tip. Plots of normalized transferred eneW/W0 for various polarizabilities of the tip are shown in Fi5. In this figure~as well as in all following figures! the mol-ecule on the left (x,0) is the donor, and the molecule on th

FIG. 5. Normalized transferred energyW/W0 for a tip scanningon thex axis at a fixed heighth511 nm. The molecular parameteare a1,res5a2,res53 nm3, g15g25109 s21, and v15v2. Theprobe polarizability@aP5(e21)/(e12)a3# is calculated witha510 nm and~a! e52.25,~b! e520.510.05i , ~c! e52110.05i ,and ~d! e52210.1i .

04542

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right (x.0) is the acceptor. The tip polarizabilities weestimated by assuming spherical tip of radiusa and complexpermittivity e:

aP5e21

e12a3. ~23!

Various permittivities allow us to model different tips, e.g.,nonmetallic tip (e.0), a metallic tip (e complex with anegative real part!, and a metallic tip, near the plasmon resnance (e'22). The result for nonmetallic tip, havingpositive polarizability, are depicted in Fig. 5~a!. The electricfield at the position of the second molecule is a superposiof the electric fields of the first molecule and the tip. Whthe tip is placed directly above one of the molecules, thtwo fields add constructively; the total field is stronger, athe energy transfer is enhanced, as demonstrated by thepeaks in Fig. 5~a!. When the tip is placed in the center btween the molecules, the interference becomes destrucand the energy transfer is suppressed. In Fig. 5~b! we showthe energy transfer for a metallic tip, whose polarizabilitynegative. The change of sign has a profound effect: the peand dips in Figs. 5~a! and 5~b! are interchanged. For a largetip polarizability one reaches a point where the energy trafers via direct and tip-mediated channels become comrable, and completely cancel each other due to destrucinterference. This is illustrated in Fig. 5~c!. If uaPu is furtherincreased, tip-mediated coupling becomes dominant.strong coupling via the tip can lead to a significant enhanment of the energy transfer, as demonstrated in Fig. 5~d!.Note that Fig. 5~d! is not symmetric, although the donor anacceptor molecules are identical. This happens becauseresonant metallic tip considerably modifies the decay ratea nearby molecule. Since the transferred energyW does notdepend symmetrically on the decay ratesG1 of emitting mol-ecules andG2 of absorbing molecules@cf. Eq. ~22!#, configu-rations with the tip placed above the donor and aboveacceptor exhibit different amounts of transferred energyW,which results in the asymmetry of Fig. 5~d!.

IV. SPHEROIDAL PROBE TIP

So far the probe tip has been treated as a pointlike dipThis simplified treatment allowed us to illustrate the bafeatures of the operation of this system of interacting ements. In order to consider effects of the finite size of thewhich will lead to the determination of the resolution of thmicroscope, we now model the probe as an elongated spoid of semimajor axisa, semiminor axisb, and complexdielectric constante. An elongated spheroid concentrates tlight intensity near the apex of the largest curvature.changing the aspect ratioa/b of such a tip, conditions ofplasmon resonances may be achieved for metals such asor silver in the visible range of the spectrum.

It is convenient to place the origin of the coordinate sytem at the center of the spheroid. The positions of two mlecular dipoles are given by displacementsr1 and r2. Weassume again that the external electric field is parallel toz axis. We use prolate spheroidal coordinatesj, h, andf.

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This choice is natural, because the surface of spheroigiven byj5j0[a/Aa22b2 in these coordinates. The coodinate transformations are

x5 f @~j221!~12h2!#1/2cosf,

y5 f @~j221!~12h2!#1/2sinf, ~24!

z5 f jh;

jP@1, ), hP@21,1#, andfP@2p,p). The two foci of theellipsoid lie on thez axis at a distance6 f [6Aa22b2 fromthe origin.

As in previous sections, we neglect retardation, and sothe problem in the electrostatic approximation. The elecstatic potential inside and outside spheroid can be wrias42

w in5 (n50

`

(m50

n

@amncos~mf!1bmnsin~mf!#Pnm~h!Pn

m~j!

~25!

and

wout5 (n50

`

(m50

n

@Amncos~mf!1Bmnsin~mf!#Pnm~h!Qn

m~j!

2E0z1m1•“1

1

R11m2•“2

1

R2. ~26!

Pnm(z) and Qn

m(z) are Legendre functions of the first ansecond kind, respectively. To find the unknown coefficieamn , bmn , Amn , andBmn , we employ boundary conditionat the surface of the spheroid. The continuity of the potenand the normal component of the electric displacemyields

w inuj5j05woutuj5j0

, e]w in

]j Uj5j0

5]wout

]j Uj5j0

. ~27!

We make use of the following expansion of 1/Rj into a seriesof harmonic functions:42

1

Rj5

1

f (n50

`

~2n11! (m50

n

~22dm0!i mF ~n2m!!

~n1m!! G2

3cos@m~f2f j !#Pnm~h j !Pn

m~h!

3H Pnm~j j !Qn

m~j!, j.j j

Pnm~j!Qn

m~j j !, j,j j ,~28!

wherej j , h j , andf j are the prolate spheroidal coordinatof the j th dipole. Substituting the above expansion into E~26! and inserting Eqs.~25! and~26! into Eq.~27!, we obtain

amnPnm~j0!5AmnQn

m~j0!2E0f j0dn,1dm,0

1@m1•cmn(1)1m2•cmn

(2)#Pnm~j0!,

04542

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.

eamnPnm8~j0!5AmnQn

m8~j0!2E0f dn,1dm,0

1@m1•cmn(1)1m2•cmn

(2)#Pnm8~j0!,

~29!

where prime (8) denotes a derivative, andcmn( j ) are vectors:

cmn( j ) 5

2n11

f~22dm0!i mF ~n2m!!

~n1m!! G2

3“ j@Pnm~h j !Qn

m~j j !cos~mf j !#. ~30!

The linear equations~29! can easily be solved, yielding

Amn5~e21!

3E0f j0dn,1dm,02Pn

m~j0!Pnm8~j0!@m1•cmn

(1)1m2•cmn(2)#

ePnm8~j0!Qn

m~j0!2Pnm~j0!Qn

m8~j0!

~31!

Similar expression can be obtained forBmn ; we only have toreplacecmn

( j ) with dmn( j ) ,

dmn( j ) 5

2n11

fe~22dm0!mi mF ~n2m!!

~n1m!! G2

“ j

3@Pnm~h j !Qn

m~j j !sin~mf j !#, ~32!

and neglect the source term proportional toE0:

Bmn52~e21!Pn

m~j0!Pnm8~j0!@m1•dmn

(1)1m2•dmn(2)#

ePnm8~j0!Qn

m~j0!2Pnm~j0!Qn

m8~j0!. ~33!

We have expressed all coefficients in terms of theknown dipole momentsmj . These dipole moments can bdetermined from a set of coupled self-consistent equatio

m15a1E1 , m25a2E2 , ~34!

whereaj is polarizability of thej th dipole, andEj is thetotalelectric field at the dipole position. This system is completanalogous to system~2! ~after elimination of the tip dipolemoment!. The electric fieldsEj can be found as gradients othe potentialwout, where the divergent term correspondingthe field of the j th dipole has to be neglected. After somalgebra, Eqs.~34! take the forms

m15a1F S z02~e21! f 2j0

3~eQ10~j0!2Q1

08~j0!j0!c01

(1)D E0

1K11m11~G121K12!m2G ,

m25a2F S z02~e21! f 2j0

3~eQ10~j0!2Q1

08~j0!j0!c01

(2)D E0

1~G211K21!m11K22m2G , ~35!

where coupling constantsK i j are given by

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K i j 5 (n50

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m~j0!Pnm8~j0!

ePnm8~j0!Qn

m~j0!2Pnm~j0!Qn

m8~j0!

3@cmn( j )

^ emn( i ) 1dmn

( j )^ fmn

( i ) #, i , j 51,2 ~36!

and

emn( j ) 5“ j@Pn

m~h j !Qnm~j j !cos~mf j !#,

~37!fmn( j ) 5“ j@Pn

m~h j !Qnm~j j !sin~mf j !#.

The symbol^ denotes a tensor product of two vectors.explicit calculations, we first determineK i j , then solve Eq.~35! for m1,2; finally, we obtain the coefficientsAmn andBmnfrom Eqs.~31! and ~33!.

The Green functionsK i j contain all information on thegeometric and material properties of the tip, relevant totwo molecules. The real and imaginary parts of diagonalementsK j j are related to the tip-induced modification of thresonant frequency and the decay rate of thej th molecule, cf.Eq. ~13!. The off-diagonal termK12 characterizes the tipmediated modification of the intermolecular coupling.

Numerical simulations have been performed to investigboth the stationary response of the system to the exteexcitation field, and energy transfer between two dipolesthe presence of the spherical probe. In what follows, thtwo processes are discussed in detail.

A. Stationary scattering

The energy radiated into the far-field zone~i.e., the signal!is proportional to the square of total dipole moment, as dcussed in Sec. II:

I scatt}um11m21mPu2. ~38!

The dipole moment of the tip can be found by analyzingasymptotic behavior of the potentialwout, and identifying theexpansion terms which correspond to the electric field opoint-like dipole:

mP5f 2

3~2A11x012B11y01A01z0!. ~39!

In what follows we discuss silver tips. The values of tdielectric constant for silver were taken from Ref. 43. Wfocused on the visible region between 500 and 600 nmthis spectral range the dielectric constant of silver chanfrom e528.310.75i at 500 nm toe5214.210.95i at600 nm, and the real part ofe linearly decreases with increasing wavelength. In our numerical simulations we csider a spheroid with the semimajor axisa550 nm and as-pect ratioa/b54. The tip plasmon resonance occurs atl'565 nm, wheree521210.85i . This probe is scannedover the molecules at a fixed vertical height of 1 nm, i.the bottom of the spheroid is 1 nm above the sample.scanning signal is plotted in Fig. 6 for three different excition wavelengths. The scattered intensity shows two distdips at the positions of the two molecules, which can thusunambiguously resolved. The molecular resonance is c

04542

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tered atl res5565 nm, and it has a linewidthDl510 nm.As can be seen in Fig. 6, the modulation of the scannsignal is most pronounced on resonance:l5565 nm. Whenthe excitation wavelength is tuned away from resonance,modulation depth of the signal decreases; see Figs. 6 an

To further analyze the influence of the excitation wavlength, we consider an experiment where the tip is fixabove one of the two molecules~say the right one!, and theexcitation wavelength is scanned through resonance.corresponding signals are shown in Fig. 8. When the molelar and plasmon resonances coincide, the signal showsingle large dip at the resonant wavelength. When theydifferent, the signal exhibits a more complicated structureclear dip is observed near the molecular resonance, ansecond dip~or shoulder! is seen near the broader plasmresonance frequency. Note that the scattering signal maenhanced whenl res is sufficiently larger than the plasmo

FIG. 7. Normalized scattered intensity for tip scanning thex axisfor excitation wavelengths varying from 530 to 580 nm. All prameters are the same as in Fig. 6.

FIG. 6. Normalized scattered intensity for a silver spheroidalmoving at fixed height 1 nm above two molecules placed onx axisat x565 nm. The plasmon resonance is at the excitation walengthl'565 nm. The molecular dipoles and excitation field aoriented along thez axis. Resonant molecular polarizabilities aa1,res5a2,res53 nm3 and the resonant molecular wavelengthl res

5565 nm. The scanning patterns are plotted for three differexcitation wavelengths:l5550 nm ~solid line!, l5565 nm~dashed line!, andl5580 nm~dot-dashed line!.

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resonance wavelength (l res5600 nm in Fig. 8!. Clearly, thehighest signal modulation can be achieved under douresonant excitation when the tip and the molecular renances coincide.

The elongated geometry of the tip confines the enhannear field to a small region and this lightning rod mechaniimproves the resolution of the system. We can charactethe resolution of the SNOM microscope quantitatively by tvisibility V(d). We assume that two identical molecules alocated on thex axis atx56d/2, and we define

V~d!5I scatt~0!2I scatt~d/2!

I scatt~0!1I scatt~d/2!, ~40!

where I scatt(0) is the intensity scattered when the tip is lcated at the center between two molecules, andI scatt(d/2) isa scattered intensity measured when the tip is positionex5d/2, i.e., exactly above the right molecule. The visibiliV(d) is plotted in Fig. 9 for three different excitation wavelengths. When the molecules are far apart, the visibilityproaches its maximum asymptotic value corresponding tovisibility of a single molecule. When the two molecules acloser together,V(d) decreases. At a certain finite intermlecular distance, the visibility reaches zero, the scatteringnal is flat in the region between the molecules, and thedips disappear.

The distanced0 whereV(d0)50 can be taken to definthe resolution of the microscope. If the molecules are ecloser, the two dips in the scattered intensity merge intsingle dip, and the two molecules look like a single objeThe definition of the visibility@Eq. ~40!# is not applicable tothis region, and we defineV(d)50 for d,d0. The visibilityis largest for resonant excitation, and decreases whenl istuned from resonance. The resolution, however, does nopend onl and in our example we findd0'4 nm.

FIG. 8. Scattered intensity as a function of the external scannexcitation wavelength. A silver tip is fixed above the right molecu(x55 nm) and the excitation wavelengthl is varied from 530 to620 nm. Results are shown for four different molecular resonwavelengths:l res5550 nm ~solid line!, 565 nm ~dashed line!,580 nm~dot-dashed line!, and 600 nm~short dashed line!. In allfour cases, the resonant molecular polarizability isa res53 nm3,and the the resonance has a widthDl510 nm.

04542

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B. Energy transfer

The methodology developed in Sec. III can be adopteddescribe the energy transfer in the presence of the sphertip. We assume that the orientations of the molecular diponj5mj /umj u, are fixed and do not change in time. In thcase, we only have to replaceaPGiPGjP with Ki j 5niK i j njin formulas~13! and ~14!. The coupling constantsKi j con-tain all information about the influence of the probe on tdynamics of two coupled dipoles. The process of enetransfer is significantly affected by the finite size of thprobe. The inhomogeneous electric fields of the molecudipoles excite higher multipoles of the spheroid, and the cpling via these higher multipole excitations strongly inflences the energy transfer. Nevertheless, some basic qutive features obtained for the pointlike dipole model of ttip remain valid.

The sensitivity of the energy transfer to the geometry onearby metallic~silver! tip is illustrated in Fig. 10, whereW/W0 is plotted for three different values of the tip asperatio a/b. In Fig. 10~a!, the destructive interference betweedirect and tip-mediated energy-transfer channels is demstrated. The suppression of the energy transfer occurs fbroad range of the tip positions. In Fig. 10~b!, the aspectratio is such that the tip has a plasmon resonance atmolecular emission frequency. The resonant coupling ofmolecules via the tip leads to a strong enhancement ofenergy transfer. Finally, in Fig. 10~c!, the tip is detuned fromplasmon resonance, and only a moderate enhancemeenergy transfer is seen.

In addition to providing an efficient coupling channel, thtip also modifies the molecular resonant frequencies andcay rates; cf. Eq.~13!. The tip-induced change of resonafrequencies can increase or decrease the detuning betwedonor and an acceptor. If the molecular resonant frequenare identical when the probe is far away, then the approaing tip can only detune the molecules from resonance.however, the molecules have slightly different resonant fquencies~e.g. due to locally different environments!, the tip

g

t

FIG. 9. Visibility of the SNOM microscope for three differenexcitation wavelengths:l5550 nm ~solid line!, l5565 nm~dashed line!, and l5580 nm ~dot-dashed line!. The molecularresonant wavelengthlR5565 nm and the resonance widthDl510 nm. All other parameters are the same as in Fig. 6.

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may tune the molecules closer to resonance, thus furtherporting the energy exchange. This effect is illustrated in F10, where we can see that the maximal enhancement oenergy transfer is larger for detuned molecules than foractly resonant molecules.

V. CONCLUSIONS

The operation of an apertureless near-field microscwas investigated under conditions of strong interactiontween the scanning tip and two nearly resonant, closelycated molecules. We showed that the observed far-fieldnal depends on the amplitudes and phases of the compolarizabilities of the tip and molecules. The interferen

FIG. 10. Normalized transferred energy vs the position osilver spheroidal tip. Molecules are placed atx565 nm. The re-sults are shown fora/b54 ~a!, a/b54.58 ~b!, and a/b55 ~c!.Solid lines correspond to resonant molecules,D22D150, anddashed lines refer to detuned molecules,D22D154. The remain-ing parameters are the same as in Fig. 6.

04542

p-.hex-

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between the dipole moments manifests itself as an enhament or suppression of the scanning signal in the far-fizone. The phase dependence makes the signal sensitimolecular detunings from optical resonance and tip detunfrom plasmon resonance, thus making the laser frequencymost sensitive tool for the control and determination of tresonant scattering properties.

We derived quantitative expressions for the modificatof the energy transfer between two closely lying resonmolecules when mediated by a tip scanning over them,showed that, under resonant conditions, this energy tranmay be enhanced by almost two orders of magnitude, duthe tip-mediated coupling channel between the molecuUnder proper conditions, destructive interference of therect and tip-mediated channels can significantly suppressenergy transfer. Thus the tip enables control of the enetransfer between the molecules. The main control mecnism is the change of intermolecular coupling. In additiothe tip also modifies the molecular decay rates and resofrequencies. All these effects act simultaneously and inence the energy transfer.

Two different models of the tip were considered. Tsimple model, in which the tip is treated as a pointlike dpole, enabled us to illustrate the basic qualitative featurethe coherent light scattering and resonant intermolecularergy transfer. The spheroidal probe model included thefects of the finite size and the elongated geometry of theThe results obtained from these two models qualitativelyhibit the same features, which do not depend on the exshape of the tip. The analytic treatment in the point dipapproximation limit, while probably oversimplified, is important and useful because it allows one to extract the unlying physics crucial for the interpretation of experimenresults.

We focused on strong tip-molecular resonant interactiOur treatment did not take into account effects due tononresonant interaction with the underlying surface ofsample. Preliminary numerical results showed that mosthe conclusions of this work remain largely unchanged wha nonresonant dielectric substrates~such as glass! are in-cluded into the scheme. However, at some special combtions of specific materials and excitation conditions, the tsurface gap may support collective localized plasmexcitations that are very sensitive to the properties ofsubstrate. The analysis of the scanning process under tconditions will be published elsewhere.

ACKNOWLEDGMENTS

The work was supported by the Tashtiot Program ofIsrael Ministry of Science, the Israel Ministry of Absorptioand the Center for Absorption of Scientists.

a

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Documents

Coherent light scattering and resonant energy transfer in an apertureless scanning near-field optical microscope