Controlling resonance energy transfer in nanostructure emitters by positioning neara mirrorDilusha Weeraddana, Malin Premaratne, Sarath D. Gunapala, and David L. Andrews

Citation: The Journal of Chemical Physics 147, 074117 (2017); doi: 10.1063/1.4998459View online: http://dx.doi.org/10.1063/1.4998459View Table of Contents: http://aip.scitation.org/toc/jcp/147/7Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 147, 074117 (2017)

Controlling resonance energy transfer in nanostructure emittersby positioning near a mirror

Dilusha Weeraddana,1,a) Malin Premaratne,1,b) Sarath D. Gunapala,2,c)

and David L. Andrews3,d)1Advanced Computing and Simulation Laboratory (AχL), Department of Electrical and Computer SystemsEngineering, Monash University, Clayton, Victoria 3800, Australia2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA3School of Chemistry, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, United Kingdom

(Received 29 May 2017; accepted 30 July 2017; published online 21 August 2017)

The ability to control light-matter interactions in quantum objects opens up many avenues for newapplications. We look at this issue within a fully quantized framework using a fundamental theoryto describe mirror-assisted resonance energy transfer (RET) in nanostructures. The process of RETcommunicates electronic excitation between suitably disposed donor and acceptor particles in closeproximity, activated by the initial excitation of the donor. Here, we demonstrate that the energy transferrate can be significantly controlled by careful positioning of the RET emitters near a mirror. The resultsdeliver equations that elicit new insights into the associated modification of virtual photon behavior,based on the quantum nature of light. In particular, our results indicate that energy transfer efficiencyin nanostructures can be explicitly expedited or suppressed by a suitably positioned neighboringmirror, depending on the relative spacing and the dimensionality of the nanostructure. Interestingly,the resonance energy transfer between emitters is observed to “switch off” abruptly under suitableconditions of the RET system. This allows one to quantitatively control RET systems in a newway. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4998459]

I. INTRODUCTION

The process of energy transfer plays an extremely impor-tant role in the optics and photophysics of solid phase phys-ical systems—specifically, those whose nanoscale structurescomprise electronically distinct components with well charac-terised absorption and emission characteristics. This process isprimarily responsible for redirecting or channeling electronicexcitation following resonance absorption. Systems exhibit-ing the effect range from those with an intrinsically complexconstitution to others created by nanoscale engineering. Inthe former connection, it is the significance and effective-ness of the transfer process in biological systems that haveled to a surge of interest in the development of nanophotonicapplications. In photosynthesis, in particular, the excitationenergy is absorbed by pigments in the photosynthetic anten-nae and subsequently transferred to a reaction center by aseries of hops between other chromophore units.1,2 In natu-rally occurring living organisms which perform photosynthe-sis, it has been found that the light-harvesting efficiency isindeed above 99%.1,3 Although this phenomenon has beenstudied extensively over many decades, there still exist openquestions regarding the underlying mechanism leading to thisremarkably high efficiency.

It has already been identified that Resonance EnergyTransfer (RET), often also known as the Forster resonance

a)Electronic mail: dilusha.weeraddana@monash.edub)Electronic mail: malin.premaratne@monash.edu.c)Electronic mail: Sarath.D.Gunapala@jpl.nasa.govd)Electronic mail: d.l.andrews@uea.ac.uk

energy transfer,4 is the main energy transfer mechanism in theprocess of energy transportation in photosynthesis.5 There-fore, conducting an in-depth research in RET and being ableto control the RET rate would be beneficial to many ofthe promising applications in nanophotonics, ranging fromphotovoltaics6–8 to bio-medical sensing9,10 where moleculardistances and interactions are examined.11,12

The process of RET occurs in the near-field regime, and itis fully quantum mechanical in nature.13–15 The inter-particleinteractions are evaluated using Quantum Electrodynamical(QED) theory due to the fact that the minimal- and multipolar-coupling Hamiltonians are many-body Hamiltonians.16,17 Thisis the one advantage of employing a fully quantized approachto study RET systems. In the framework of QED, couplingbetween bodies is mediated by the exchange of virtual pho-tons, which are not detected. Their lifetime is limited by thetime-energy uncertainty principle manifesting in one or moreemission-absorption processes.18–20

Generally, RET is the dominant energy transfer mecha-nism between emitters (a donor and an acceptor) in nanometerproximity, where, typical of the Forster theory, the rate has acharacteristic R�6 distance dependence (R here is the separa-tion between donor and acceptor molecules).4,21 Other meansto control a RET system are the spectral properties of thedonor and acceptor (donor’s emission spectrum and accep-tor’s absorption spectrum) or by the relative orientations ofthe transition dipole moments.19,22,23 In fact, it is also possibleto control the RET rate purely by means of the surround-ing environment while leaving the RET pair geometricallyand chemically unchanged, by a passive third-body24,25 or

0021-9606/2017/147(7)/074117/10/$30.00 147, 074117-1 Published by AIP Publishing.

074117-2 Weeraddana et al. J. Chem. Phys. 147, 074117 (2017)

by an auxiliary laser beam impinging on the donor-acceptorsystem.26–28 Further, in cavity QED, inter-particle interactionscan be modified due to both the propagating and evanescentmodes inside the cavity.29–34

However, there are ongoing debates stimulated by mod-ern nanofabrication techniques, about controlling RET purelyby means of the nanophotonic environment. Indeed, theoryand experiments have revealed both enhanced and inhibitedRET rates for many different nanophotonic systems, rangingfrom plasmonic systems to spasers.35–37 Recently, Wubs andVos studied the Forster resonance energy transfer in nanopho-tonic media with weak dispersion,38 by considering results forthe analytical model of a pair of dipole emitters in the vicin-ity of an ideal mirror. Even though basic principles of RETand control of the RET rate in basic building blocks such asatoms and molecules have been extensively studied, there stillexist open questions regarding the mechanisms of controllingthe RET phenomenon in nanostructures such as quantum dots(QDs) and nanowires (NWs). As the quantum confinementin nanostructure materials makes them capable of provid-ing unique and superior optical properties,39–41 in this study,we specifically focus on RET in nanostructures of differentdimensions.

Thus, the main purpose of the present article is to pro-vide new theoretical insights into controlling the rate of RETin nanostructure systems, purely by means of the environ-ment while leaving the RET pair geometrically and chemicallyuntouched. To this end, we have developed a simple modelbased on the QED framework, by involving the placement ofan ideal mirror near the RET nanostructure emitters. We seekto discover what particular effects arise in the matrix elementand the transfer rate when input excitation is located in thevicinity of a single mirror, in terms of atomic transition fre-quency and relative distances, constructing a detailed pictureof how individual virtual photon behaviour is modified by thevicinal non-absorbing mirror.

II. QED FRAMEWORK FOR MOLECULARINTERACTIONS IN THE VICINITY OF A MIRRORA. Theory

We begin by introducing a suitable quantum frameworkto describe the quantum level light-matter interaction forRET. Within the QED framework, the light-matter interac-tion is studied using perturbation theory, to accurately cast thecoupling matrix element.13,42 The experimentally measurablequantity, Γtrans, the energy transfer rate of any photophysi-cal process proceeding from any initial system state I to afinal state F, can then be determined through Fermi’s goldenrule,43

Γtran =2π}|Md

FI |2ρ, (1)

in which ρ represents the density of the final system states.The coupling strength between I and F is characterized by thequantum amplitude (coupling matrix element, resonant dipole-dipole interaction), MFI , formally cast as an infinite series,which is expressed as follows:

MFI =

∞∑q=1

〈F |Hint(ξ)

(Hint(ξ)EI −H0

)q−1

|I〉 , (2)

where EI is the initial system energy and H0 represents theunperturbed system Hamiltonian. The parameter q denotes thepower of the radiation-matter interaction Hamiltonian, Hint(ξ),which comprises contributions for each species ξ locatedat Rξ ,

Hint(ξ) =∑ξ

−µ(ξ) · E(Rξ ). (3)

In the above expression, µ(ξ) is the electric-dipole momentoperator and the electric field operator is given by E(Rξ ). Fur-ther, in the framework of molecular QED, the Hamiltonianfor the radiation and matter system can generally be writtenas13,18

Htotal =∑ξ

Hint(ξ) +∑ξ

Hmol(ξ) + Hrad . (4)

The operator Hmol(ξ) is the molecular Hamiltonian in the non-relativistic Born-Oppenheimer approximation, and Hrad is thesecond-quantized radiation field Hamiltonian.44

The eigenstates, |τ 〉, of a basis Hamiltonian given in Eq.(4) form a composite set that can be expressed by

|τ〉 = |matτ〉 |radτ〉 ≡ |matτ ; radτ〉 . (5)

|matτ〉 in Eq. (5) defines the status of all particles (or molecules,according to the system), comprising a product of state vectorsfor each matter, and |radτ〉 is the radiation (number) state. Thecase of direct resonance energy transfer near a single mirroris depicted in Fig. 1. Here, the virtual photon can be createdeither at a donor or an acceptor. Therefore, with two virtualphoton-matter interactions and the corresponding interactionHamiltonian acting as a perturbation, the quantum amplitudecan be calculated from the second term (q = 2) of the expansiongiven in Eq. (2),

MFI =〈F |Hint |R1

1〉 〈R11 |Hint |I〉

EI − ER11

+〈F |Hint |R2

1〉 〈R21 |Hint |I〉

EI − ER21

, (6)

where I and F are the initial and final states, respectively, andR denotes the intermediate state.

B. Mirror-assisted RET

It is at this stage that the effect of including the mirrorcomes into play. We shall assume that the mirror is located at z= 0 and perpendicular to the z axis. The electric field operatorin the z > 0 region will be quantized as29,45

E(m,m′)(Rξ ) =∑p,λ

(}cp

2V ε0

)1/2 [a(λ)(p)E(m,m′)(Rξ )

− a†(λ)(p)E∗(m,m′)(Rξ )]

, (7)

where V is an arbitrary quantization volume and a(λ)(p) anda†(λ)(p) are the familiar annihilation and creation operatorsthat, respectively, modify the number of photons of modes;m = p, λ and m′ = p′, λ. p, p′ are the incident and reflectedwave vectors, respectively, and λ is the polarization. Withineach pair of braces, the paired terms effectively represent field

074117-3 Weeraddana et al. J. Chem. Phys. 147, 074117 (2017)

FIG. 1. (a) Schematic depiction of theresonant energy transfer for nanoparti-cles in front of a single mirror, whereD′ is the associated image of a donorparticle (D); the arrow in each nanopar-ticle signifies the dipole moment. (b)The two Feynman diagrams for reso-nance energy transfer. Here, 0 denotesthe ground-state, α and β denote theexcited levels for the donor and accep-tor, respectively.

expansions for the two separate half-spaces delineated by themirror.

Therefore, the interaction matrix element can be derivedfrom Eq. (6),

〈R11 |Hint |I〉 = 〈1(p, λ); D0A0 | − µ(D).E(m,m′)(RD)

− µ(A).E(m,m′)(RA)|DαA0; 0〉

= i∑

m

(}cp

2V ε0

)1/2

µ0αi (D)E∗(m,m′)i(RD), (8)

where i, j are the Cartesian coordinates,

〈F |Hint |R11〉 = −i

∑m,m′

(}cp

2V ε0

)1/2

µβ0j (A)E(m,m′)j(RA),

〈R21 |Hint |I〉 = i

∑m,m′

(}cp

2V ε0

)1/2

µβ0i (A)E∗(m,m′)i(RA),

〈F |Hint |R21〉 = −i

∑m,m′

(}cp

2V ε0

)1/2

µ0αj (D)E(m,m′)j(RD).

(9)

By the application of Eqs. (8) and (9) in Eq. (6) gives thegeneral formula for the quantum amplitude for RET,

MFI =µ0α

i (D)µβ0j (A)

2V ε0

∑m,m′

E∗(m,m′)i(RD)E(m,m′)j(RA)p

k − p

−E(m,m′)i(RD)E∗(m,m′)j(RA)p

k + p. (10)

In the above expression, a concise notation for the transitiondipole moments is introduced, e.g., µ0α(D) ≡〈D0 |µ(D)|Dα〉,and p is the corresponding photon wave number that need notbe equal to k.

C. Remarks on the proposed QED model

It is important to highlight the fact that in our model,the light-matter interactions between the donor and acceptorhave been considered within a perturbative quantum electro-dynamics framework.29 Imposing quantum conditions on theclassical electromagnetic (EM) field is the key step in quantumelectrodynamics. In many instances, this is clearly seen by theappearance of reduced Planck’s constant, }, in expressions.Nevertheless, the final expression of the coupling matrix ele-ment, which is derived from the second order perturbation, isfree from }. This is because the } terms in the coupling matrix

element cancel each other out.16 Therefore, this should notbe misinterpreted as a classical result.46 The final expressionfor the resonance energy transfer essentially contains } fromFermi’s golden rule in Eq. (1).13

III. MIRROR ASSISTED RESONANCE ENERGYTRANSFER IN NANOSTRUCTURES

It is fascinating to investigate the near field energy transferbased on QED under man-made alterations, enforced by con-finement to regions bounded by surfaces like mirrors. Sucheffects are now becoming important in practice, primarilyin quantum-optics experiments on particles in cavities thatare bounded in various degrees.33,47,48 Therefore, we con-sider a system composed of two nanoparticles separated bya distance R (center-to-center separation) and a single mirror(z = 0) as illustrated in Fig. 1. In this section, we analyt-ically explore the effects on the process of energy transferdue to a mirror located in the vicinity of donor and acceptornanostructures.

Now, let us consider donor and acceptor nanoparticles inthe xz plane, and the coordinates are, respectively, RD = (0, 0,zD) and RA = (Rx, 0, zD + R). Rx is the distance to the acceptorin the x direction from the z axis, and R is the center-to-centerseparation distance between donor and acceptor particles. Asingle mirror in the x, y plane is placed at z = 0, as illustratedin Fig. 1(a); zD is the distance to the donor from the mirrorin the z direction. The corresponding Feynman diagrams areillustrated in Fig. 1(b).

A. RET: Quantum dot to quantum dot in the vicinityof a mirror

Here, a pair of QDs separated by distance R interactingwith a single mirror is considered. Thus, from Eq. (7), by con-sidering electromagnetic waves in either side of the mirror,E(m,m′)(Rξ ) = e(λ)(p)eip·Rξ −e(λ)(p′)eip′ ·Rξ . Here, e(λ)(p) is thepolarization unit vector [e(λ)(p) being its complex conjugate],now owing to the spherical symmetry,

p = p(sin(θ) cos(φ), sin(θ) sin(φ), cos(θ)), (11)

p′ = p(sin(θ) cos(φ), sin(θ) sin(φ),− cos(θ)). (12)

The wave-vector and polarization summations can be eval-uated by following the standard techniques of Craig andThirunamachandran13 [i.e., choosing the orthogonal frame set

074117-4 Weeraddana et al. J. Chem. Phys. 147, 074117 (2017)

e(1)(p), e(2)(p) and p as the independent frame givese(1)

i (p)e(1)j (p) + e(2)

i (p)e(2)j (p) + pipj = δij]. Finally, extending

the boundaries of the quantization volume, each point in thep-space represents a realizable p-vector and the wave vectorconverts to an integral as V −→ ∞,

limV→∞

1V

∑p

≡

∫d3p

(2π)3. (13)

Now converting to the spherical coordinates,∑p

⇒V

4π3

∫ ∞0

p2dp∫ π/2

0sin(θ)dθ

∫ 2π

0dφ. (14)

Next, directly substituting into Eq. (10) and convertingthe discrete summation over the virtual photon wave vectorgive18,29

M(QD) =1

16π3ε0

∫ ∞0

dp∫ π/2

0dθ

∫ 2π

0dφ p3 sin(θ)

∑λ=1,2

µ0αi (D)µβ0

j (A)

(k2 − p2)

[e(λ)

i (p)e(λ)j (p)

{k(eip(Rx sin(θ) cos(φ)+R cos(θ))

− e−ip(Rx sin(θ) cos(φ)+R cos(θ))) + p(eip(Rx sin(θ) cos(φ)+R cos(θ)) − e−ip(Rx sin(θ) cos(φ)+R cos(θ)))}

+ e(λ)i (p′)e(λ)

j (p′){k(eip(Rx sin(θ) cos(φ)−R cos(θ))− e−ip(Rx sin(θ) cos(φ)−R cos(θ))) + p

(eip(Rx sin(θ) cos(φ)−R cos(θ))

− e−ip(Rx sin(θ) cos(φ)−R cos(θ)))}+ e(λ)i (p)e(λ)

j (p′){k(− eip(Rx sin(θ) cos(φ)−(R+2zD) cos(θ))

+ e−ip(Rx sin(θ) cos(φ)−(R+2zD) cos(θ))) + p(−eip(Rx sin(θ) cos(φ)−(R+2zD) cos(θ)) + e−ip(Rx sin(θ) cos(φ)−(R+2zD) cos(θ)))}

+ e(λ)i (p′)e(λ)

j (p){k(− eip(Rx sin(θ) cos(φ)+(R+2zD) cos(θ))+ e−ip(Rx sin(θ) cos(φ)+(R+2zD) cos(θ)))

+ p(−eip(Rx sin(θ) cos(φ)+(R+2zD) cos(θ)) − e−ip(Rx sin(θ) cos(φ)+(R+2zD) cos(θ)))}]

. (15)

Performing the contour integration and applying the residue theorem, when Rx = 0, yield

M(QD) =µ0α

i (D)µβ0j (A)

4πε0

[− k2(δij − RiRj)

( cos(kR)R

−cos(k(R + 2zD))

(R + 2zD)

)+ k(δij − 3RiRj)

( sin(kR)

R2−

sin(k(R + 2zD))

(R + 2zD)2

)+ (δij − 3RiRj)

( cos(kR)

R3−

cos(k(R + 2zD))

(R + 2zD)3

)+ i

{− k2(δij − RiRj)

( sin(kR)R

−sin(k(R + 2zD))

(R + 2zD)

)− k(δij − 3RiRj)

( cos(kR)

R2−

cos(k(R + 2zD))

(R + 2zD)2

)+ (δij − 3RiRj)

( sin(kR)

R3−

sin(k(R + 2zD))

(R + 2zD)3

)}]. (16)

On the examination of Eq. (16), it is evident that the thirdterm (in both real and imaginary components) is dominant inthe short-range (kR � 1), and the first term is dominant inthe long-range (kR � 1). Moreover, it can be shown that thecoupling matrix element is fully transverse with respect to Rin the long-range, whilst for the shorter-range, it contains bothtransverse and longitudinal components.18

RET occurs in the near-zone region. Hence, we willexplicitly focus on the short-range for both QD and NW cases.

Thus, by applying kR � 1 for the near-field zone, and ifµ(D), µ(A) are in the positive x direction,

Mxx(QD) =µx(D)0αµ

β0x (A)

4πε0

[{ 1

R3−

cos(2kzD)

(R + 2zD)3

}

− i{ sin(2kzD)

(R + 2zD)3

}]. (17)

Moreover, the magnitude of the coupling matrix elementcan be expressed as

|Mxx(QD)| =|µx(D)0α | |µ

β0x (A)|

4πε0

×

√( 1

R6−

2 cos(2kzD)

R3(R + 2zD)3+

1

(R + 2zD)6

). (18)

The plots of the normalized coupling matrix element areshown in Fig. 2. In the development of the plots for Sec. III,the following values were used for dipole moments:49 |µ0α(D)|= |µβ0(A)| = 5×10−30 C m. Figure 2(b) shows the normalizedRDDI (resonant dipole-dipole interaction) for various valuesof the separation distance of the mirror and the donor quantumdot (zD). For the smaller separation distances, the RDDI withthe presence of a mirror is lower than the RDDI of the mirror-less case.

1. QD-QD: Mirror-less case

When the mirror-donor separation distance is large(zD→∞), relative RDDI (M/M0) reaches 1, reducing the resultof Eq. (16) to the no-mirror expression given as

limzD→∞

M(QD) =µ0α

i (D)µβ0j (A)

4πε0R3

{(δij − 3RiRj)

(cos(kR)

+ kR sin(kR))− (δij − RiRj)(k

2R2 cos(kR))}

− i{(δij − 3RiRj)(sin(kR) − kR cos(kR))

− (δij − RiRj)(k2R2 sin(kR))

}. (19)

This is exactly the same result obtained for direct couplingbetween two quantum dots when there is no mirror placed

074117-5 Weeraddana et al. J. Chem. Phys. 147, 074117 (2017)

FIG. 2. Normalized RDDI strengthsfor two quantum dots parallel to the mir-ror: (a) schematic for the coupling ofQD-QD near a mirror, (b) the normal-ized coupling matrix element |M |/|M0 |as a function of the distance from thedonor quantum dot, D, to the mirror,zD, where the particle separation R = 10nm, (c) the normalized coupling matrixelement |M |/|M0 | as a function of thedonor-acceptor separation distance, R,for five different zD (3 nm, 5 nm, 15nm, 20 nm, and 25 nm), and (d) nor-malized RDDI strengths as a functionof the wave number corresponding to theresonant atomic transition frequency forthree different cases (zD = 3.5 nm, 4 nm,and 5 nm).

nearby.21,25,50 Therefore it can be deduced from the aboveresults that by applying the limits (zD→∞), we can success-fully simplify the mirror-assisted RET to the no-mirror case(direct RET between two QDs).

Figures 2(c) and 2(d) depict the variation of normalizedRDDI strengths with respect to the donor-acceptor separationdistance (R) and photon wave number corresponding to theatomic transition frequency (k) for various zD values. As shownin Fig. 2(c), for larger zD values, an enhancement in the cou-pling matrix element compared with the mirror-less RDDI canbe observed. However, a reduction in the coupling betweendonor-acceptor particles can be seen for smaller zD values.Interestingly, regardless of the position of the mirror, whenthe donor and acceptor are located in very close proximity, theinfluence of the mirror becomes negligible.

Moreover, the normalized coupling matrix element oscil-lates with the wave number, as illustrated in Fig. 2(d).

Furthermore, when the mirror is moved closer to the quantumdots, higher energy enhancement or more deterioration can beachieved depending on the suitably chosen k value.

Substitution of Eq. (17) into the Fermi golden ruleexpression yields

Γtrans(QD) =|µx(D)0α |2 |µ

β0x (A)|2ρ

8π~ε20

×( 1

R6−

2 cos(2kzD)

R3(R + 2zD)3+

1

(R + 2zD)6

). (20)

We illustrate the normalized energy transfer rates for threedifferent energy shifting cases as shown in Fig. 3. Figure 3(a)shows the RET rate enhancement factor as a percentage withrespect to the donor-acceptor separation distance, for threemirror to donor distances (zD = 8 nm, 15 nm, and 20 nm).It can be observed that the normalized energy transfer rate

FIG. 3. Normalized energy transfer rates for three different energy shifting cases: (a) RET rate enhancement factor (as a percentage) in QDs as a function ofthe donor-acceptor separation distance, for three mirror to donor distances (zD = 8 nm, 15 nm, and 20 nm), (b) RET rate deterioration factor (as a percentage) inQDs as a function of the donor-acceptor separation distance, for three different mirror to donor distances (zD = 2 nm, 5 nm, and 7 nm), and (c) switching action:the coupling matrix element as a function of zD.

074117-6 Weeraddana et al. J. Chem. Phys. 147, 074117 (2017)

gradually increases with R. Furthermore, the energy transfercan also be diminished for certain mirror-particle separationdistances as depicted in Fig. 3(b).

Moreover, the RET can be completely “cutoff” in theregion highlighted in Fig. 3(c) owing to the fact that the electricfield component [see Eq. (7)] is modified by the light reflectedfrom the mirror surface. In Fig. 3(c), the center-center separa-tion distance of donor and acceptor QDs and the wave numberare R = 10 nm and k = 1 × 108 m�1, respectively. The switch-ing region varies according to the values of these parameters.This can be understood by careful examination of Eq. (18).Here, when zD is small, one of the additional terms introducedby the reflected waves from the mirror ( 2 cos(2kzD)

R3(R+2zD)3 ) becomes

more dominant and makes the coupling matrix element 0 ornegligible. When zD increases, a sudden upturn of the RDDIcan be observed. This is due to the high influence of smallerzD values on the RDDI, for a suitably oriented QD pair. Thishigh impact of zD on the RDDI gradually decreases to 0 withincreasing zD [applying zD → ∞ limits on Eq. (17)]. This dras-tic upturn opens up potential applications for highly sensitivepositioning tools.51–53

B. RET: Nanowire to nanowire in the vicinity of a mirror

In this section, the process of RET in a system consistingof idealized 2D parallel nanowires of length L is considered.Owing to the cylindrical symmetry of NWs, it is conve-nient to model EM waves using a Hankel function of ordern.25,54,55

In a similar manner to the previous case in Sec. III A,if µ(D), µ(A) are oriented in the positive x direction and twonanowires are located along the z axis (Rx = 0) as shown inFig. 4(a), then we have

Mxx(NW ) =µ0α

x (D)µβ0x (A)k2

4Lε0

{Y0(kR) − Y0(kR + 2kzD)

− i(J0(kR) − J0(kR + 2kzD))}. (21)

Similarly, the magnitude of the quantum amplitude in thepositive x direction can be expressed as follows:

|Mxx(NW )| =|µ0α

x (D)| |µβ0x (A)|k2

4Lε0

√{Y2

0 (kR) + J20 (kR)

+ Y20 (kR + 2kzD) + J2

0 (kR + 2kzD)

− 2Y0(kR)Y0(kR + 2kzD)

− 2J0(kR)J0(kR + 2kzD)}. (22)

We depict the normalized coupling matrix element forNW-NW in the vicinity of a mirror as shown in Fig. 4. Fig-ure 4(b) shows the normalized RDDI for various values of theseparation distance (zD) of the mirror and the donor NW. Forthe smaller separation distances, the RDDI with the mirrorin the vicinity is lower than the RDDI of the mirror-less case.The normalized coupling matrix element fluctuates around thevalue of 1 and gradually decreases with zD.

1. NW-NW: Mirror-less case

Similar to the QD case, for higher mirror-donor separationdistances (zD → ∞), relative RDDI reaches 1, reducing theresult of Eq. (21) to the mirror-less expression given as

limzD→∞

Mxx(NW ) =µ0α

x (D)µβ0x (A)k2

4Lε0{Y0(kR) − i(J0(kR))}.

(23)

The above expression can also be obtained by derivingthe direct coupling between NW to NW, from the secondorder perturbation theory.25,55 Therefore, by imposing the lim-its (zD → ∞), we can successfully simplify the mirror-assistedRET to the no-mirror case (direct RET).

The magnitude of the quantum amplitude in the absenceof the mirror becomes

|M0(NW )| =|µ0α(D)| |µβ0(A)|k2

4Lε0

√{Y2

0 (kR) + J20 (kR)

}.

(24)

FIG. 4. Normalized RDDI strengthsfor two nanowires parallel to the mir-ror: (a) schematic for the coupling ofNW-NW in the vicinity of a mirror, (b)the normalized coupling matrix element|M |/|M0 | as a function of the distancefrom the donor nanowire, D, to the mir-ror, zD, where the particle separation R= 10 nm, (c) the normalized couplingmatrix element |M |/|M0 | as a function ofthe donor-acceptor separation distance,R, for five different values of zD (2 nm,4 nm, 6 nm, 10 nm, and 15 nm), and (d)normalized RDDI strengths as a func-tion of the wave number correspond-ing to the resonant atomic transitionfrequency for three different cases (zD= 4 nm, 5 nm, and 10 nm).

074117-7 Weeraddana et al. J. Chem. Phys. 147, 074117 (2017)

As expected, owing to the photon behaviour in the cylin-drical symmetry and the modified electric field component dueto the mirror surface, the coupling between two nanowiresmodified to Eq. (22) from Eq. (24).

Furthermore, Figs. 4(c) and 4(d) show the variation ofnormalized RDDI strengths with respect to R and photon wavenumber corresponding to the atomic transition frequency (k)for various zD values. As shown in Fig. 4(c), for larger zD

values, an enhancement in the coupling matrix element com-pared with the mirror-less RDDI can be observed. However,a reduction in the coupling between donor-acceptor particles

can be seen for smaller zD values. Similar to the case of QDs,regardless of the position of the mirror, when the donor andacceptor are located in very close proximity, the influence ofthe mirror becomes negligible, converging the normalized fac-tor to 1. In addition, unlike in the variation of the normalizedmatrix element with k in QDs, same amount of enhancementand reduction of the quantum amplitude can be observed fordifferent zD values in NWs due to the behaviour of the virtualphoton propagation in a 2D realm [see Fig. 4(d)]. To deter-mine the rate of transfer, Fermi’s rule given in Eq. (1) can beused,

Γtrans(NW ) =|µ0α

x (D)|2 |µβ0x (A)|2k4πρ

8L2~ε20

{Y2

0 (kR) + J20 (kR) + Y2

0 (kR + 2kzD) + J20 (kR + 2kzD)

− 2Y0(kR)Y0(kR + 2kzD) − 2J0(kR)J0(kR + 2kzD)}, (25)

Γtrans(NW )Γtrans,0(NW )

=Y2

0 (kR) + J20 (kR) + Y2

0 (kR + 2kzD) + J20 (kR + 2kzD) − 2Y0(kR)Y0(kR + 2kzD) − 2J0(kR)J0(kR + 2kzD)

Y20 (kR) + J2

0 (kR). (26)

The plots of rate enhancement and reduction factors forEq. (26) are shown in Figs. 5(a) and 5(b) for various values ofthe D-A distance, R. These figures exhibit similar patterns tothose observed for QDs in Sec. III A. In fact,∼130% incrementof the energy transfer can be gained when the mirror is placed15 nm away from the donor NW. Additionally, up to 80%transfer inhibition can be obtained when zD = 2 nm.

C. Design guidelines for mirror-assisted RET

In Secs. III A and III B, we have developed a fundamen-tal theory for describing the mirror-assisted resonance energytransfer in dimensionally constrained nanostructures (QDs andNWs) within the framework of quantum electrodynamics. Wehave also shown that transfer rates can be robustly enhancedor suppressed by positioning a mirror near a donor-acceptorpair under favourable conditions.

Therefore, in this section, we will elaborate designguidelines for mirror-assisted RET with suitable modelparameters. The following values were used for generationof plots:49 |µ0α(D)| = |µβ0(A)| = 5 × 10−30 Cm and k = 1

× 108 m�1. Moreover for all the plots generated, we used thedonor-acceptor separation distance of R less than 10 nm. How-ever, we exclude the results for R < 1 nm as those signifythe possibility of wave function overlap (Dexter zone).56 Adetailed explanation about this region is given in Sec. IV.

1. RET enhancement: Higher RET rate enhancements canbe achieved in NWs than in QDs when placed in thevicinity of an ideal mirror.

• Quantum dots: Fig. 3(a) illustrates the energytransfer enhancement with respect to the donor-acceptor separation distance. Up to 13% enhance-ment can be obtained when zD ≤ 15nm.

• Nanowires: Approximately 130% increment ofthe energy transfer can be gained when the mirroris placed 15 nm away from the donor NW. This isshown in Fig. 5(a).

2. RET suppression: The RET rate can be successfullyinhibited up to a considerable extent in both QDs andNWs.

FIG. 5. Normalized energy transfer rates for two dif-ferent energy shifting cases: (a) RET rate enhancementfactor (as a percentage) in NWs as a function of donor-acceptor separation distance, for three mirror to donordistances (zD = 8 nm, 10 nm, and 15 nm), (b) RET ratedeterioration factor as a percentage in NWs as a functionof donor-acceptor separation distance, for two differentmirror to donor distances (zD = 2 nm and 5 nm).

074117-8 Weeraddana et al. J. Chem. Phys. 147, 074117 (2017)

• Quantum dots: When the donor-mirror separationdistance is less than 7 nm, the mirror-assisted RETrate starts going below Γ0 (rate corresponding tothe no-mirror case). This is because, when zD issmall, one of the additional terms introduced bythe reflected waves from the mirror ( 2 cos(2kzD)

R3(R+2zD)3 )

becomes more dominant making the couplingmatrix element smaller than that of the no-mirrorcase. Thus, more than 90% of the rate reductioncan be achieved when zD = 2 nm as depicted inFig. 3(b).

• Nanowires: Similar to the QDs, approximatelywhen zD < 6 nm, RET rate suppression can beobtained. This is shown in Fig. 5(b) for three dif-ferent zD values. Nearly 83% of the rate reductioncan be observed when zD = 2 nm.

3. Switching action: This can be observed only in QDs, asillustrated in Fig. 3(c). A sudden up turn can be observedwhen the mirror is placed very close to the donor QD(i.e., ∼1.6 nm, when R = 10 nm).

IV. DISCUSSION

A fundamental theory has been developed for describingthe mirror assisted resonance energy transfer in dimensionallyconstrained nanostructures within the framework of QED. Wehave performed analytical derivations and plotted the couplingmatrix elements and energy transfer rates. The results havedemonstrated the effect of the mirror in the vicinity of twodonor-acceptor nanostructures, according to the relative dis-tances among each quantum object along with the variationof quantized EM modes; these are the factors that determinethe controllability of the resonance energy transfer rate. Thiscan be understood by the modification to the electric field29,45

given in Eq. (7), which contributes additional components tothe coupling matrix elements as derived in Eqs. (10) and (21)for QDs and NWs, respectively. This quantum level elementof interference allows a high degree of control of the RET rate(RET rate enhancement, suppression, and switching actions)between nanostructures.

Figure 2 characterizes the normalized RDDI factor ofQD-QD as a function of the donor distance to the mirror, donor-acceptor distance, and wave number (corresponding to theresonant atomic transition frequency, fr = kc/2π). The rangewhere the coupling between donor and acceptor QDs is effec-tively controlled by the distance to the mirror is zD < 40 nm.This range is set by the donor-acceptor distance of 20 nm. How-ever, if the donor-acceptor distance is set to the most typicalRET separation distance in the order of 10 nm, then the preciseRET controlling range becomes zD < 20 nm. This agrees withthe results of Ref. 38.

Therefore, we have studied the coupling between donorand acceptor QDs as a function of R and k, for various valuesof zD, in the range of zD < 20 nm. Interestingly, when 8 nm< zD < 20 nm, the rate enhancement up to 13% can beachieved [see Figs. 2(c) and 3(a)]. In addition to this, whenzD < 8 nm, the RET rate can be suppressed to a certain extent(90%) as shown in Figs. 2(c) and 3(b). Furthermore, the drastic

upturn shown in Fig. 3(c) opens up potential applications forhighly sensitive positioning tools.51–53

In Sec. III B, similar results exhibit by the couplingbetween NW-NW near a single mirror. Nevertheless, in con-trast to the QD case, a more than 120% increment in the energytransfer rate can be gained as shown in the plots of Fig. 5(a).

In contrast to the emitter configuration used in this paper,the work reported in Ref. 38 employed both parallel and per-pendicular configurations of dipoles with respect to the mirror.Compared to the classical approach used in Ref. 38, the presentpaper formulated a full quantum electrodynamical treatmentfor mirror assisted energy migration with the aid of Feynmandiagram methods. Interestingly, as in our case, the above arti-cle shows that the RET rate can be inhibited or enhanced bycarefully positioning a mirror close to particles, typically a fewnm. In Ref. 38, higher enhancement can be observed comparedwith the rate inhibition for dipole emitters. Nevertheless, inour case, QDs exhibit higher percentages of inhibitions thanenhancements, while NWs gain higher RET rate enhancementsthan inhibitions, when located near a mirror. Moreover, in thepresent article, we could achieve higher rate enhancements forNWs than QDs while both nanostructures show markedly rateinhibitions.

Note that values of R < 1 nm generally signify the pos-sibility of wave function overlap. Here, another mechanismof energy migration occurs, called the Dexter mechanism.56

When the particles are almost touching each other and theinteraction between them is high, there is a possibility thatas the excited electron is transferred from the donor particleto the empty higher energy level of the acceptor particle, theelectron in the ground state of the particle is simultaneouslymoved to fill the “electron hole” of the donor. The net physicalresult is the same as in the Forster transfer, but here the energytransfers follow an exponential rate formula. Nonetheless, inthe kinds of nanoscale systems of widest interest, the donorand acceptor components are in general designedly separatedby a distance sufficiently large to preclude any significantdegree of wave function overlap. Therefore, in this article, weexclude the Dexter zone to focus specifically on the Forsterrange.

It is immediately clear that the far-zone and intermediate-zone terms in Eq. (16) exhibit similar patterns to the near-zoneresults. However, it provides different coupling matrix elementvalues due to the k2 and k terms in the numerator. In the long-range, the photons have field components that are transversewith respect to the displacement vector, opposed to the near-field case, where both transverse and longitudinal componentsplay vital roles in the process of energy migration.16 Neverthe-less, the cases where the mirror positioned far-away from thedonor-acceptor system in the far- and intermediate-zone pro-vide results exactly similar to the respective no-mirror cases.This proves that our results are valid for all three domains:near-, intermediate-, and far-field.

V. CONCLUSION AND OUTLOOK

Tailoring the quantum level light-matter interaction withthe nanophotonic environment provides accurate control overthe resonance energy transfer between two nanostructure

074117-9 Weeraddana et al. J. Chem. Phys. 147, 074117 (2017)

particles. Despite the wide applications of RET innanosciences, using nanophotonics to control RET hasremained a debated and complex challenge over many decades.Here we have demonstrated the exquisitely controlled energytransfer within a single donor-acceptor nanoparticle pairplaced in close proximity of a mirror, at nanoscale dis-tances comparable to the separation between the particles. Afull quantum electrodynamical treatment for mirror-assistedenergy migration has been formulated with the aid of Feynmandiagrams and perturbational theory.57,58

It is interesting to note that other aspects of photophysicsbeing markedly modified in the proximity of a mirror have alsorecently received attention in connection with chiral speciesand circular polarisations.59–62

Summarizing, the ensuing results demonstrated the possi-bility of altering the strength of the resonance energy transferbetween two nanoparticles by careful engineering of the spac-ing and inclusion of an ideal quantum mirror in the vicinity.Therefore, the analysis demonstrates a mechanism to inhibitor markedly enhance the transfer of energy between discretecomponents, potentially in any multi-quantum dot/nanowiresystem, inviting surface and layer applications.

As future extensions of our work, it will be interestingto study the different orientations of the nanostructures withrespect to the mirror. Moreover, a possible optical switchingaction with a laser beam impinging on the particle systemrepresents scope for potential development of the theory.

In conclusion, we believe that our new insight into thecontrol of the near-field resonance energy transfer offers con-siderable new scope to be exploited for accurately measuringmolecular separations, for the manipulation of quantum infor-mation, and for much efficient bio-sensors, and holds potentialfor improved photovoltaics.

ACKNOWLEDGMENTS

The work of D.W. is supported by the Monash UniversityInstitute of Graduate Research. The work of M.P. is supportedby the Australian Research Council, through its DiscoveryGrant No. DP140100883.

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