Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array
S.V. Lobanov, T. Weiss, D. Dregely, H. Giessen, N.A. Gippius, S.G. Tikhodeev
aa r X i v : . [ phy s i c s . op ti c s ] O c t Emission properties of an oscillating point dipole from a gold Yagi-Uda nanoantenna array
S. V. Lobanov , , T. Weiss , D. Dregely , H. Giessen , N. A. Gippius , , and S. G. Tikhodeev , M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russia A. M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilova Street 38, Moscow 119991, Russia Max Planck Institute for the Science of Light, G¨unter-Scharowsky-Straße 1Bau 24, D-91058 Erlangen, Germany th Physics Institute and Research Centers Scope and Simtech, University of Stuttgart, D-70550 Stuttgart, Germany LASMEA, University Blaise Pascal, 24 Avenue des Landais, F-63177 Aubi`ere Cedex, France (Dated: 30 September 2011)We investigate numerically the interaction of an oscillating point dipole with a periodic array of optical Yagi-Uda nanoantennas in the weak coupling limit. A very strong near-field enhancement of the dipole emission bythe resonant plasmon mode in the feed element is predicted in this structure. It is shown that the enhancementstrength depends strongly on the dipole position, the direction of the dipole moment, and the oscillation fre-quency. The radiative intensity of the point dipole from appropriate places next to one feed element may exceedthe radiative intensity of an equivalent dipole in free-space by a factor of hundred. In spite of only one directorused in each nanoantenna of the array, the far-field emission pattern is highly directed. The radiative e ffi ciency(the ratio of the radiative to the full emission) appears to be around 20%. PACS numbers: 78.67.-n, 42.60.Da, 73.20.Mf
Nanophotonics has been in the focus of intensive inves-tigations in recent years. One of its numerous areas is thenanoscale controlling of light emission from a single moleculeor quantum dot. Promising tools for the realization of this goalare optical nanoantennas [1–13].Metal antennas are traditionally used for controlling the ra-diation pattern of electromagnetic wave emission in the radioand microwave frequency range. Though the electromagneticproperties of metals in the optical range di ff er significantlyfrom that in the radio and microwave range, it seems to bereasonable to use the main concepts of radio antennas in theoptical range as well. It has been suggested [2, 3, 5, 7, 13]to construct a nanooptical antenna with elements that are ar-ranged as in the radio Yagi-Uda antennas.Yagi-Uda antenna consists usually of one or two reflectors,one feed element and several directors with appropriately se-lected scattering phases (reflector and director are slightly de-tuned inductively and capacitively). As it has been recentlyshown, the nanoantenna elements can be nanorods [3, 5, 13],core-shell [2, 7] or spherical [12] nanoparticles. All elementsscatter the light and the resulting interference forms a highlydirected beam along the antenna axis. The size of the elementsin such optical Yagi-Uda nanoantennas have to be smallerthan the light emission wavelength in free-space. Such op-tical nanoantennas only work e ffi ciently in narrow frequencydomains, where the interaction of the emitter with light is res-onantly enhanced because of the excitation of so-called local-ized plasmons [14, 15] in the nanoantenna elements.Spontaneous emission is not an intrinsic atomic property,but it depends sensitively on the local density of photonicmodes at certain frequencies in a microcavity [16, 17], or,equivalently, on the local electromagnetic field value at theposition of the quantum emitter [4, 6, 11]. Using resonances,it is possible to increase the local electromagnetic field sig-nificantly, and, resultantly, to enhance and redirect the dipoleemission. In the case of localized plasmon resonances, thecollective excitation of electrons at the plasmon frequency leads to a considerably enhanced emission rate when the pointdipole is located in the vicinity of metallic nanoparticles withthe appropriate orientation of its dipole moment [5, 8, 10].The exact description of photon radiation from the quantumemitter located in some metal-dielectric environment is verycomplicated. A convenient approximation is a model of anoscillating point dipole. It oscillates with constant frequencyand magnitude fixed by the external source (so-called weakcoupling limit). In other words, the emission of a current ~ j ( ~ r , t ) = ~ j · δ ( ~ r − ~ r ) · e − i ω t inside an environment with spa-tially modulated permittivity has to be calculated. This systemis now classical and can be described by Maxwell’s equations.The goal of this paper is to calculate the radiation patternand emission rate of one oscillating point dipole located in theperiodic array of optical Yagi-Uda nanoantennas. Each an-tenna in the array consists of three rectangular gold elements,director, feed, and reflector. The point emitter is located a fewnanometers from the edge of one feed element. To the bestof our knowledge, the emission properties of an oscillatingpoint dipole from such a structure have not been investigatedyet. The so far investigated structures have been either singleYagi-Uda nanoantennas [2, 3, 5, 7, 13], or arrays of simplerspherical shapes [8]. Using arrays of antennas in combina-tion with several emitters with controlled phase di ff erence ofoscillation, we can control the emission directivity addition-ally [18], as in phased antenna arrays.The calculation of the radiation characteristics from oscil-lating dipoles in two-dimensional photonic crystal slabs canbe performed using the method of scattering matrix [19, 20].Its main idea is to divide the structure into two parts adjacentto a plane through the dipole, to find the scattering matrices ofthese parts, and to combine them by considering the bound-ary conditions at the plane with the dipole. We use the Fouriermodal method (FMM) [19, 21], which can be improved by theformulation of the correct Fourier factorization rules [22] andadaptive spatial resolution [23], as well as of matched coordi- Z XY
FIG. 1: (Color online) Lateral view of a periodic array of opticalYagi-Uda nanoantennas with an x -oriented oscillating point dipole(black point with arrow) located at a horizontal distance of 5 nmfrom the feed element of one nanoantenna. nates for more complex metallic shapes [24].The structure of interest is shown schematically in Fig. 1.It consists of a glass superstrate ( ε = . ε = . x - and y -axis are equal to 450 nm and 300 nm, respec-tively. Each antenna consists of three rectangular gold paral-lelepipeds of 30 nm height and 100 nm width. The lengthsof the top (director), middle (feed), and bottom (reflector) el-ements are 220, 250 and 300 nm, respectively. They are lo-cated in glass and the vertical distance between them is equalto 100 nm. We assumed the gold permittivity to be describedby the Drude formula with 9016 meV plasma frequency and81 meV damping rate.Our first goal is to calculate the directional pattern as wellas the emission spectra in the direction normal to the antenna
500 1000 1500050100150200250 Energy (meV) N o r m a li z ed i n t en s i t y P top P bottom FIG. 2: (Color online) Calculated spectra of the emission in top (bluecurve) and bottom (green curve) directions of the oscillating pointdipole (directed along x ) coupled to the gold Yagi-Uda nanoantennaarray. The intensity is normalized to the maximum radiation inten-sity of an equivalent dipole (with the same magnitude and the samefrequency) in free space. plane. It makes sense to normalize the computed emissionintensity P ( ϑ, ϕ ), i. e., the Poynting vector of the dipole emis-sion in the far field as a function of the spherical angles ϑ and φ , to the maximum intensity of the emission of a point dipolein free-space, which oscillates with the same magnitude andfrequency. Thus, we can easily distinguish the enhancementof emission ( P >
1) from the attenuation ( P <
1) compared tothe dipole located in homogeneous vacuum.To characterize quantitatively the full dipole emission andits radiative part, it is convenient to define the Purcell factorand its radiative part as F P = (cid:8) Σ ( ~ P · d ~ A ) (cid:8) Σ ( ~ P · d ~ A ) , and F radP = (cid:8) Σ ( ~ P · d ~ A ) (cid:8) Σ ( ~ P · d ~ A ) , (1)respectively. Here, Σ and Σ denote spheres with an infinitelysmall and large radius, respectively, that surround the pointdipole. ~ P is the Poynting vector of the dipole emission fromthe antenna array, ~ P is its counterpart for dipole emission infree space. The antenna absorption losses can be characterizedby the non-radiative part of the Purcell factor F nrP , F nrP = F P − F radP . (2)Now, we can also introduce the radiative e ffi ciency η indicat-ing the radiative part of dipole emission, η = F radP F P . (3)In what follows the results of these quantities will be presentedemploying 1633 spatial harmonics in the FMM [21, 24] .The calculated emission spectra in top ( + z ) and bottom ( − z )directions of the x -directed oscillating point dipole located in-side the periodic array of gold Yagi-Uda nanoantennas areshown in Fig. 2. The dipole is placed on the horizontal sym-metry axis along x -direction at a distance of 5 nm to the edgeof the feed element (see in Fig. 1). Only one strong and narrowresonance occurs in the emission spectra in the top directionat the photon energy ~ ω =
820 meV ( λ = . µ m). Its mag-nitude is about 250 and the FWHM is 107 meV. The emissionto the bottom direction is significantly smaller. It has even aminimum at the plasmon resonance.Figure 3 depicts the calculated radiation pattern of emis-sion P ( ϑ, ϕ ) ~ e r at the resonant photon energy ~ ω =
820 meV(where ~ e r is the radial unit vector of a spherical coordinatesystem centered in the point dipole position). The calculatedradiative part of the Purcell factor according to the secondEq. (1) appears to be as high as 80 (see also Fig. 4 below),which indicates a very strong enhancement of the dipole emis-sion by the feed element.In spite of only one director employed in our Yagi-Uda an-tenna, the emission in top direction appears to be highly di-rectional. In order to characterize it, the angular directivity D ( ϑ, ϕ ) can be calculated [25], which indicates the part of the FIG. 3: (Color online) Calculated 2D polar diagrams (solid lines) of far-field emission in xz (left panel) and yz (central panel) planes, as wellas the full 3D-directional diagram (right panel) for the x -oriented oscillating point dipole coupled to the gold Yagi-Uda nanoantenna array atresonant photon energy ( ~ ω =
820 meV). The emission intensity is normalized to the maximum radiation intensity of the equivalent dipole(with the same magnitude and the same frequency) in free space and shown in the right panel as colored surface P ( ϑ, ϕ ) ~ e r . The dipole positionis in the center-of-coordinates. Dashed lines in left and central panels show the scaled by a factor of 1 / P ( ~ d , ϑ, ϕ ), theemission of a pair of synchronized dipoles separated by distance ~ d (see explanation in the text). full emission radiated along the direction ( ϑ , ϕ ), D ( ϑ, ϕ ) = π P ( ϑ, ϕ ) ! P ( ϑ, ϕ ) d Ω . (4)Its maximum value D max = max[ D ( ϑ, ϕ )] is called directiv-ity and indicates the antenna’s ability to form a narrow beam.The larger the directivity, the narrower is the light beam. Anisotropic radiator would have a directivity of 1, whereas foran oscillating point dipole in free space, D max = .
5. In thecase of the radiation pattern of Fig. 3, D max = .
7, i.e., it ex-ceeds the directivity of the dipole in free space by a factor of3.1. In [5] the directivity of a single Yagi-Uda antenna withthree directors has been calculated as 6.4, which is only 1.37times larger than in our case with only a single director. Fur-thermore, we can also increase the directivity using more thanone dipole emitter coupled to di ff erent antenna array elementswhich is shown below.It is also instructive to investigate the dependence of theemission enhancement on the dipole position and on the ori-entation of its dipole moment respectively to the feed ele-ment. Figure 4 shows the calculated dependence of the ra-diative part of the Purcell factor F radP with respect to the dis-tance between the feed element and the dipole for x -, y -, and z -directed dipoles. The emission enhancement of the antennaarray decreases with increasing distance. The strong polariza-tion and distance dependence demonstrates the local nature ofthe antenna-dipole enhancement.For the di ff erent distances indicated in the inset of Fig. 4,the emission is only significantly enhanced for the x -polarizeddipole, which means that a plasmon mode with charges oscil-lating along x is excited in the system. Such mode leads to avery strong enhancement of emission and determines the di-rectional pattern. As the electromagnetic field near the edges of the feed element is known to have a dipolar character atthe fundamental plasmon mode, it is not surprising that theradiation pattern of the z -polarized dipole displaced verticallyby 10 nm above the edge of the feed element (see the insetin Fig. 5) nearly coincides with that of the x -polarized dipole5 nm apart from the edge of the feed element (see Fig. 3).Unlike a z -polarized dipole in free space, the majority of theemission is directed along the dipole polarization (i.e., alongthe z -axis). The magnitude, however, is about 3 times smallerthan that of the x-oriented dipole.In this point let us discuss the results of the full Purcellfactor F P and the radiative e ffi ciency η . The calculated re-sults are shown in Fig. 5, where the dipole is placed 10 nmabove the feed (i.e., in a homogeneous transparent glass). ThePurcell factor is rather large and reaches the maximum of F P ≈ −
300 when the dipole is located above the edgeof the feed element. Our numerical analysis shows that thegrowing oscillations of the calculated Purcell factor when the x − coordinate approaches the edges of the director, feed, andreflector elements (vertical dashed lines in Fig. 5) are the ar-tifacts of our calculation method. The radiative e ffi ciency ap-pears to be almost independent of the dipole’s x -coordinateand is about 20% when the emitter is above the feed element.It should be mentioned that we omitted the calculation ofthe full Purcell factor as well as the radiative e ffi ciency η inFig. 4, where the dipole is placed inside a modulated layer. Inthis case, it appears that we cannot calculate these quantitiescorrectly in the FMM, because the discontinuous permittivityfunction of such a layer is described by a truncated Fourierexpansion. Thus, due to the Gibbs phenomenon, also the non-absorptive surrounding of the metal exhibits a small imaginarypart. The point dipole approximation, however, fails if thedipole is placed inside a lossy material.Finally, we would like to mention that the antenna arrayallows to control the emission directivity further, by usingseveral emitting dipoles in di ff erent positions and control-ling their phase di ff erence of oscillations. Figure 3 showsthe scaled by 1 / ~ d , proportional to integer numbers ofperiods (see dotted and dashed lines in the left and centralpanels). The emission intensity of the synchronized pair ofdipoles normalized to the maximum emission intensity oftwo non-synchronized dipoles is then simply P ( ~ d , ϑ, ϕ ) = P ( ϑ, ϕ ) h + cos (cid:16) π n ( ~ d · ~ e r ) /λ (cid:17)i , where P ( ϑ, ϕ ) is the direc-tional pattern of a single dipole (solid line), and n is the re-fractive index of superstrate. Note that the polar diagrams P of the synchronized dipoles demonstrate the e ff ect of super-radiance: the Purcell factor of two synchronized closely posi-tioned dipoles becomes twice larger than unsynchronized. Itis seen also that it becomes possible to increase the directivityusing two emitting dipoles.To conclude, we show that the gold Yagi-Uda nanoantennaarray enhances and simultaneously directs radiation of the os-cillating point dipole. The dipole-nanoantenna enhancementdepends very strongly on the oscillating frequency, dipole po-sition, and orientation of its moment. It becomes possible tocontrol the emission directivity further by using several syn-chronized emitting dipoles attached to di ff erent antennas inthe array. This opens a way to manipulate an excited-statelifetime of a quantum emitter and to fabricate narrow beam-ing nanoscale antennas in the optical range.We acknowledge support from BMBF (13N 9155, 13N10146), DFG (FOR 557, FOR 730, GI 269 / / UFA,
FIG. 4: (Color online) Calculated dependencies of the radiative partof the Purcell factor F radP on the horizontal distance between thedipole and the edge of the feed element for a x -, y -, and z -polarizeddipole (squares, circles and diamonds, respectively). The verticaldashed line marks the position exactly above the edge of the reflec-tor. The geometry is explained in the inset: the x -polarized dipoleis shown as a black dot with arrow, it is is centered with respect tothe feed element along y - and z -direction and shifted in x -direction,along the dotted horizontal line. FIG. 5: (Color online) Calculated dependencies of the Purcell factor F P (squares) and the radiative e ffi ciency η (diamonds) on the dipoleposition for a z -polarized dipole. The x -coordinate of the dipole ischanged from −